fr
AN ABTRCT
OF THE THESIS OF
(Naie)
Date Thes
(Dcgree)
(Major)
---------- _-rt
---------
-S-.--
_____________
The
(Major Professor)
purpose of
thesis
to study the behavior of the well- knovm Eeriaitian polynomieJs and related functions in the
calculus.
That , instead of an in.uinite interval we concern ourselves only with a finite niiber of equally spaced points. There are many
between the
and the infinitesimal calculus;
Is not surprising
we have succeeded in findIng new functions of the same type as those
develoed by Hernito in 1864.
The ordinary Hermitien
developed by dIfferentiatIng e
2
We develop the new functions by taking the differences of the binomial fune-
under
( 's_.
I
1, where t
the order of difference and s is the
consideration. In
manner we obtain the H-polinowia1s and the point
which correspond to thoce functions of the sanie notation in the ordinary calculus.
These functions turn out to possess the all-important property of biorthogonality with respect to summation from
O to n. Difference equations have been developed and extensive tables calculated.
For a prao;ical application of the theory developed we have succeeded in removIng a great deal of the labor involved In the graduation of frequency
Such work formerly required the attention of an expert bx type
B
have reduced the process to mere multiplication and
Furthermore, we have found new methods of ordinary
addition. the

BIORTHOGONAL FUNCTIONS FOR FREQUENCY
DI STRIBTJTIONS by
WIILIA1i MATTEESON STONE
A THESIS submitted to the
OREGON STATE COLLEGE in partial fulfillment of the requirements for the degree of
MASTER OF ARTS
June 1940

APPROVED:
Red
Head of the Department of Mathematics
In Charge of Major
1/
Chairran of School Graduate Coirunittee
Chairman of State College Graduate Council

ACTLEDGE}.ENT
The
wishes to acimowledge his indebtedness to
Professor
VT.
E. Mime, who suggested the problem and directed the work at every stage.

TABLE OF CONTENTS
Chapter
1.
Title Page
I1TRODUCTION ....................................
. . i
ALYTICAL
TI
ONS
IVELOPMEW OF BIORTHOGONAL FUNC- i. The H and O Functions ...................
.
2
2. Determination of Coefficients ...............
.
4 li.
111.
3.Differenoequations .........................
5
4.BiorthogonalProperty. ................
NUMERICAL DELOPMENT
OF BIORTHOGONAL FUNCTIONS l.NunierioaiDifferenoes ...............
2.RecursionFornulas ..........
APPROXIMATION OF FREQUENCY FUNCTIONS
1. The P(x,$) Function
14
2
.
Table3 of P(x,
) .
.
. . .
..........
.
........
.
.16
Examples ................................
1V. CURVE FITTING BY POLYNOMIALS
1.
Minimization of the Re3idues .................
2
.
Exanple
.
........
. .
.............. .23 i
Table A ............................. ........26
2 Table B ........
.
........................ ....40
3 e
Table C. ................ .......... ....... ..83

BIORTHOGONAL FUNCTIONS FOR FREQUENCY DISTRIBUTIONS
INTRODUCTION
One of the fundamental problems encountered by statisticians is the graduation of frequency curves. Vfith present methods this is far from simple if there is lack of synnnetry inherent in the curve. The usual procedure in such cases is to employ the well-knorL Gram-Charlier type A distribution, vhich necessitates a change of axes, the calcula- tion of 2nd, 3rd, and 4th moments, and, finally, the use of extensive tables of the derivatives of the error function.*
By following closely the established theory, but confining our- selves to the study of n+l equally spaced points, we have succeeded in reducing this labor by an airiazing degree. Equipped with the tables in the appendix of this paper a person of very mediocre matheiriatical ability may easily treat cases which formerly required the attention of an expert. And there is no loss in accuracy; if we care to construct the tables we can obtain 8th or 10th degree approximation.
A second application of the theory herein developed has been found in the problem of curve-fitting by polynomials. However, this new methods offers no substantial advantage over others that have been known for centuries.
*Fishor, Arne. Mathematical Theory of Probability.
MacMillan. Co., 1930. Second edition.
New York, The

2
1. A1'IALPICAL DEVEIP1ENT OF BIORTHOGO1QA.L
FU1'ICT
IONS
The H and Fimotions
In the ordinary calculus the well-knovin Hermitian poly-noiriials and related functions are defined throuhout an infinite interval. To study the behavior of these functions in the finite calculus we start with the condition that the interval shall consist of n + i equally spaced points. This introduces a new parameter, n; it must be clearly understood at this stase that n is invariant throughout all of the operations and developments of this paper.
For convenience we reduce the constant difference between points to unity by some such linear relationship as
X X0 + ha, where he is a constant and s is the new variable.
The functions to be investigated are defined as
(1) H5(t)
(n -
).
The finite difference operator, /\
, is defined as
= (s + i) -
In
(i) t is the order of the difference, s is the particular point considered, and n + i is the number of points in the interval. The operand conforms to the usual binomial notation that indicates
(n-t' (n-t)! s-tJ
(s-t)1(n-s)!

3
Of course these factorials have meaning only if s is a positive integer.
If such is not the case we must make use of the Ganmm function.
It is easily shovin that the t-th difference of a function of s may be expressed as t
At eu(s)
-
- k=O
(-1)
(t
\ k) u(s + t -
(2)
If we substitnte for the operator in (i) we have
H8(t) t k=O
(-1)k t
(k,
(n -
- t\ k).
The stmimation in (2) may not be extended beyond k s. Otherwise the
(n - factor
( - k) becomes zero. Hence we have developed a factorial polynomial in t of degree s.
()
By altering the factorials in
(2) we find a second form, t s
(t) = k'O
¡n - s'\ (l)k()( k)
('\
' fn'
.
Lt)
By interchanging s and t in (2) we introduce a new function, de- fined as
(4) s k "S\ (n- s\ =y k) t kuO
- k).
It i-s evident that in (3) and
(4) the summations will always be carried to the sanie number of terms. Comparing the two we see at once that
(5) () H3(t)
) t
This relationship is of considerable importance in the application of

the theory developed in this paper to the calculation of theoretical frequencies.
4
Determination of Coefficients
We have. sho that H8(t) is a factorial polynomial. Hence we may write
(6) i15(t) = a0 + a1t
(1)
+
(2) a2t + ......... +
(n) at where t is defined as t(k)
- t(t
- i)(t
-(t
- k
+ 1).
The coefficients in (6) are quickly evaluated by giving t suoces- sive integral values and using the smation form of H3(t). We Lind that a0= al = -
2s
--
(n\ sJ a2 -
22 s(s - lJ
1n\
12 n(n - 1) (ss) and that, in general, ak =
(_1)k 2k
_ l)(s - 2) k n(n - l)(n -
)
"""(s
- k + 1) fn\
'"(n
- k + i) 'ks)
(7)
Usiri the factorial notation we may now write (6) as
H5(t) =
.8)
>!i k-O
(1)k k (k) (k)
2 s t k! (k)

5
Obviously this smmation will terminate when k is equal to either s or t. If s is given a partioulrr value in (7) becomes a factorial polynomial in t of degree s.
If t is held oonstent while s varies we have a similar expression for t
(8) t() ()
(_1)k 2k t(k) k (k)
8
(k)
It i8 interesting to note that
(a) is of the form,
Et(s) which may be considered analagous to the fundamental relationship of the ordinary Hermitian functions.
Difference Equations
Early investigators of the Hermitian functions in the infinites- mal calculus encountered the pair of adjoint differential equations,
:_1
+ dx
X
43 dx
+ (n + l)u
= O
,
_y dx2
-xY
+ ny o. dx
By similar methods we have found a pair of difference equations which are satisfied by the H and functions respectively. In the functional form these are vritten as
(9) (n - t)u8(t + 1) + (2s - n)Hs(t) + tH8(t - 1) - O

(1°) (s + l)t
(s + 1) + (2t
-n)t
(s) + (n +
1-
)4t(
- a Q.
As might be expected, (9) and (io) are both linear difference equations with rational coefficients. Comparison of a fornai solution of (9) with (7) will furnish conclusive proof that the difference equa- tian is really satisfied by
115(t).
To obtain such a solution we shall use the method of operators as published by MiineThonpson.* For convenience we shall drop the sub- script, and, to reduce (9) to a more standard form, we shall set t + i a and then drop the primes. Equation (9) is now written as
(li) (n + i - t)u(t) + (2s - n)H(t
- i) + (t - i)u(t - 2) - o.
Following the notation of Mime-Thompson we introduce a new opera- tor, t:?, defined as
(12) (x) = r(
- r + 1) Emu(x) fl(x - r - m + i) where r is a fixed number, m is an arbitrary number, and the operator,
E, is in turn defined as
E-m u(x) a u(x - m).
*Milne-Thompson, L. M. The Calculus of Finite Differences. Greenwich,
England, MaeMillan & Co., 1933. First edition, pp. 434-477.

In our problem we shall take r - O.
By successive applications of the f operator we find that
7
- t H(t
- 1) and that
2 fH(t) =t(t-1)H(t-2)
By zmiltiplying (U) through by t and substituting the operators we can reduce the original difference equation to the operation equation,
(13) [t(n + i - t) + (2s
-n)+jH(t)
- O.
We shall now introduce a new operator, if
, defined as lTu(x) (x -
(14)
(x - r) [u(x) - u(x -
Since we have taken r O we can easily show that
(15)
7T+
.
Suoh a polynomial operator as x may be expressed, in general, as
F(x). From a wei1-knon theorem* we see that if
F(x) u(x) - F(Tr +°
) u(x) then
(16)
F(x)u(x)f(1T)+f1()°+f2(jr)f+.
............
2! u(x)
*Milne-Thompson, L. M. The Calculus of Finite Differences. Greenwich,
England, MacMillan & Co, 1933. First edition, pp. 440.

[]
To evaluate the polynomial operators in (13) we apply (16). After making some obvious reductions we obtain a comparatively simple equation in the operational form:
(17)
[(n
+ + 2(- + s + i) ] n(t) - o
Vie now make the asswption that
(18) H(t) = fm so that (17) may be vritten as
(19) [7T(n + i
-
)
+
2(-+ s + i) f]
Using the we1l-iown theorem that, in general, n
= fk(m)f we may write the indicial equation as
(20) m(n+1-m) 0.
The two solutions of (20) are just what we should expect, 8ince the difference equation was of the second order. However, we shall consider only the case where m
- 0. The other value of n is integral and leads to zero factors in the denominators of the coefficients of the factorial serios solution. The theory of difference equations has been extended to treat such cases, but a second solution has no fur-
*Milne-Thompson, L. M. The Calculus of Finite Differences. Greenwich,
England, MacMillan & Co., 1933. First Edition, pp. 442.

ther bearing on our problem.
The recurrence formula for the determination of the coefficients is easily shocn to be
(21) k(n+l-k)ak +2(-k+s+l)ak_l
O.
Initial value of ak, corresponding to k - O, is arbitrary. By giving k successive integral values we find that the k-th coefficient is
= a0
(_1)k 2k
- 1) k! n(n
- 1)
(s - k + i)
(n
- k + 1)
If we set a o
¡n.
-I and substitute for the operators in the infinite series,we find a par- ticular solution of
(9) to be
L'
(22) H8(t) =
I
I)
(..1)k 2k 8(k)
_________ k
I
(k) t(k)
This expression for
118(t) agrees exactly with that found on page 4.
Proof that (lo) is satisfied by
4t() is quite similar.
Biorthogonal Property
By a very coirnnon treatment of the difference equations we may prove that H5(t) and
+t() are biorthogonal with respect to swmna- tion from O to n. In both (9) and (lo) we shall introduce a now inde- pendent variable, x. If we multiply (9) by
4t(x) and (io) by
118(x) and subtract the new forms we shall have all terms of the new equation

i surnable with respect to x. Stmming from O to n we have
(23) 2(s
- t)
- t(x)Hs(X) -
(n -X)t(X)H5(X
+ 1) +
(x + l)t(x + l)H3(x) n xt(x)H8(x
- 1) n x-O
(n + i -X)t(X-l)H5(X).
If s and t are not equal we may perform this surnmat ion at once by giv- i successive integral values from O to n. It is easily seen that in any set of four terms we will have two cancelled by the preceding set and two by the following set. Hence the right hand side of (23) re- duces to the two terms.
(' -n)t(n)H5(m
+ 1)
- (n + l)(n
+ l)H8(n).
The first of these is zero at once; the second also reduces to zero because the factor,
(n\
) , is zero when s is negative or greater than n. Equation (23) may now be written as
(24) t(x)H8(x) - o. s t
When the s and t subscripts are equal we may combine (7) and (8) and write n
(23') x-O
5(x)H8(x)
(n n
'\s) x-O
'-s
kO k k (k)
(-1) 2 s k
(k)
(k
X
2
A rigorous proof has not been found as yet btrt we will asstrie

11 that the sumnation with respect to x does not depend on s. By means of numerical tables developed in the next chapter this assumption may be verified for all possible values of s in a set of n + 1 points.
Setting s = O the problem reduces at once to the swmnnation with respect to x of
.
The ooinplete form of (24) will then be
(24') n x=O
Ot(x)Hs(x)
(O if st. l2n1f s=t
Hence H5(t) and
O.t(s) are biorthogorial with respect to sumama- tion of the independent variables from O to n. As in the ordinary cal- culus this property is of prime importance in the development of the various types of approximating functions.

12
1]..
NUMERICAL DEVELOPMENT OF BIORTHOGONAL
Numerical Differences
(i)
Vie have shown that the H polynomials are defined by
H5(t)
At nL2
(n-t
- t
In this chapter we shall develop several important properties by giv-. n and t particular values and taking the numerical differences.
(n
-
As a beginning let us find the n-th difference of dently the binomial operand reduces to
(O\
Evi-
-
)
/
A table of these diff ences up to n n
8 is quite easily constructed. t
1 2 3 4 5 6 7 8
-
-6
'21

13
For t = n
- i our operand reduces to
(
Ì1\
)
. t
(1
"s
1 2 3 4 5 6 7 8 i i -7 i -6 i
-5 20 i -4 14 i -3 9 -28
1 -2
5 -14 i -i 2 -5 14 o o o o i -1 2 -5 14
-1
2 -5 14 i -3
9 -28
-1 4 -14 i
-5 20
-i 6 i -7
-1
1
By preparing a number of these tables we are able to construct tables of the t-th differences of the function,
(
(n - t\ t) where s and t take integral values from O to n. For a specific case let us set n = 11 we shall get a square array of nunibers, twelve on a side. The numbers in the s-th column will satisfy a polynomial in t of degree s, namely H8(t). Similarly the t-th row of ntwibers will satisfy t() defined by (8).
Since both H8(t) and pt(s) have the factor,
(n) we see that neither function is defined if s '1
O or s n. But if we relax the restriction that t must be a positive integer between
O and n we may extend the table indefinitely for negative or positive values of t.
Furthermore, it may be easily demonstrated that the polynomial, ii8(t),

14 has s zero points, all of which are real and lie in the interval be- tween O and n.
For examples of these tables, hereafter designated as type A, the reader is referred to the appendix at the end of this paper.
Recursion Formulas
We have show. that H8(t) satisfies the recursion formula,
(9) (n - t)H3(t + i) + (2s - n)115(t) + t H5(t
- 1) and that
Ot(s) satisfies the formula,
0
(lo)
(s + 1)4t(s + i) + (2t
- )t() + (n +
-s)(s
- 1) 0.
From the type A tables we have constructed we can verify that any square block of four adjoining cells will satisfy the ainszixìg1y simple recursion formula,
(25) H(s,t) - H(s+l,t) + H(s,t+l) + H(s+l,t+l)
0, where H(s,t) indicates the cell common to the s-th column and the t-th row. The importance of (25) cannot be underestimated. If we set s and t equal to zero the first three terms may be found at one, re- gardless of the valuo of n, so that repetition of the process affords a quick and easy method of table construction. And by taking advantage of the obvious symetry the labor is reduced to a minimum.

15 li].
APPROXThtATION OF FREQUENCY FUNCTIONS
The P(x,$) Function
Let us assume that a frequency function, f(s), may be represented by a terminating series of the factorial function, °() That is,
(26) f(s)
&t where t is the order of the difference and n+]. is the nuniber of points under consideration.
If we multiply (26) through by 113(t), change to dummy variables, and sum both sides with respect to s from O to n, we get, after changing the order of the terms, n n
(27) f(s) u8(t) k ll() n to
=
The last step in (27) follows at once from the orthogonal property.
Solving for the coefficient we find that
(28) s-t = I
2n s=O f(s) H5(t)
Obviously, (26) may now be written as n
(29) f(s) 2_
2n f(s) n
I t=O
11(t) 4(s)
By setting x a s we find an entirely navi formula for the approx-

16 ixnation of a theoretical frequency function,
(30) f(x) = f(s) >1 H5(t) (x)
We here introthoe some suitable notation for the stmnnation
H(t) j(x) such as Pn(x,$) It turns out that the sun'nnation may proceed only as far as t = n - 1, as the n-th term reduces to zero. Equation (30) may now be written as
(31) f(x) = n
I f(s)
P(x,$)
2fl s=0 where the subscript
an integer between
O and n - 1. This means that the sunm.ation is stopped when t p,
80 we may get nearly any desired degree of accuracy by taking p fairly large.
Tables of P(x,c)
To be of practical value our new function must be made up into tables of convenient form. To derive a formnla to facilitate the construction of these tables, hereafter designated as type B, let us make use of the difference equation satisfied by H5(t). By changing the notation a little we may write (9) as
(32) (2s -
= - t ti()
-
(n - t) t+i()
Setting X = S,
(32) may be written as

17
(33)
(2x
- n) q(x) -t th1(x) -
(n
- t) t+i(X)
If we multiply (32) by t(x) and
(33) by shall get and subtract we
(34) 2(s
= t_1(x) t(s) + (n
- t) t+1(x) j(s)
-t t() s)
-
(n
- t) t(x)
To reduoe the left hand side of (34) to the surrunation form of we divide through by the factor, 2(s - x), nmdtiply both sides by the expression,
!
(n
- t (n
- t)!' substitute by means of (5), and, finally, stun both sides with respect to t. Equation (34) is now of the form,
P(x,$) = s
2 p
(n
(s
- )!
- x) t0
(tlx t() t+l(X)
(t-i) (n-t)) t t()
(n-t-1)/
(35)
- t(x) t_l(s)
- t(x) ti()
(t-i)
!
(n-t)7 t
L
(n-t-1) f
By giving t successive integral values it is easily seen that each set of ternis cancels half of the preceding set. This means
the stumnation
stopped at
= p we shall have
(36)
P(x,$) s/ (n-s)!
2(s-x) p!(n-p-i)J p+l(x) (s)
-
(x)

18
Since the c functions on the right are quickly found from table A,
(36) gives us a perfectly general method for calculation of table B. A
A slight difficulty arises when s - x, as
P(xs) is then an indetermn- ant form,
.
We may get around this, however, by taking advantage of the fact that the numbers in a vertical column will satisfy a polynoini.al in s of degree p. If we can calculate enough cells directly from (36) a reversal of the difference operation will carry us over the indeterm- ina.ut points.
A simple check on the accuracy of any type B table is available.
If in the fundamental equation,
(31) f(x) = -
2 n
[ f(s)
P(xs) s=O we set f(x) f(s) we shall then have
(37)
1 n
(x1s)
2r
With a calculating machine this test may be applied very quickly.
Examples
To illustrate the power and simplicity of our new method of treat- ing frequency functions we have incorporated two examples. It is evi- dent that besides a great reduction in labor we have actually gained in accuracy.
The fol)owing table gives the frequency distribution of the heighth

of 346 men.*
The class interval has been reduced to unity by some appropriate linear relationship. We have 9 points so we will use table B, n = 8, a p = 5. For purposes of comparison the results obtained by the use of the proba'oility integral have been included.
19
3
4
5 s
0
1
2
6
7
D
'J f(s)
1
2
9
48
131
102
40
13
A
-X
P-funo. f(x)
1.2
1.7
7.0
55.0
122.1
106.4
40.7
11.6
A
Z
Integral f(x)
.1
1.2
12.0
55.1
114.2
108.4
45.4
8.7
To compare our results with those obtained by the use of the
Grain-
Charlier series we shall treat the following frequency distribution Of pensioned workers of an American corporation.** Thirteen different groups are considered so we use the table for n - 12, p - 5.
*Ricler, Paul R.
Louis, Mo.,
An Introduction to Modern Statistical Methods. Saint
Johm Wiley & sons, 1939. First edition, pp. 75.
**Fiaher, Arne. Mathematical Theory of Probability.
MacMillan Co., 1930. Second edition, pp. 264.
New York, The

2
3
4
5
6
7
8 s o i
9
10 ii
12 f(s)
1
6
17
48
118
224
286
248
128
38
13
2
1 f(x)
- P.
.9
5.8
18.5
47.8
116.].
223.7
294.2
241.3
124.5
41.5
10.6
3.3
.7 f(x) -G.C.
2
291
241
126
44
15
3
5
17
48
118
219
1

2].
(38) iV. CURVE FITTflTG BY POLYNOMIALS
Minimization of the Residues
To fit a polynomial to a set of n+]. points let us assume that we have a function, say T(t), defined as
T(t) a5u5(t)
We have sho'wn that H3(t) is a polynomial in t of degree s. The values of these polynomials as t takes on integral values may be found from table A.
To irsure the best fit we must make the sun of the squares of the residues a minimi.n, which is quite analogous to the process of curve- fitting by such we1l-knoin functions as the Legendre polynomials.
In this case, however, we must introduce a weight function,
Ç order to establish the orthogonality of the H polynomials.
)
Since() is sumraable, nowhere negative, and different from zero over a given set of n+l points it fulfills the conditions laid down by Jackson.
*
(39)
The sun to be nünimized is n
S = n
()
[ft
-
T(t)
12 j
Substituting for T(t) we may write this as n
(40) s - r
(n\ t)
L f(t) - n a n3(t)
J
2
Equating to zero the derivative of
S with respect to the i-th coeffic-
*Jackson, D,.mham. American Mathematical Society Colloquium Publica- tiens, Vol. Xl, pp. 95. 1929.

dent we have
(41) t0
() f(t) 111(t)
- jj t0
() u1(t)
Y a
;(t)
= o s0
22
The polynomials are orthogonal with respect to the weight function.
That is,
(42) n
:iii
(fl t-O
(O if i
# s.
H(t)
118(t) -
3
2' (
\sI if i
-
Applying (42) and solving for the coefficients we may now write
(43) f(t) =
I
2n s
I f(t) us(t)2
H5(t)
(s
Setting t = x and expanding the simmtion we may express the fitted curve, f(x), as
(44) f(x) =
(n
2 i
()
Jyl
( f(t) H it-O \
0(t)
(
+
1
() c(t) Hl(t)}
111(x)
+ ............ .......... ............. b
......
+
(U
2n()
\n
(t.o
() f(t) ;(t)
Çx)
Coefficionts may be found by the use of table A. To facilitate this type of curve-fitting we have included in the appendix some of the
Hermitiari polynomials up to the fifth d.gree.

As a simple illustration of the application of this method we shall fit a fifth dcree polynomial to the following set of points:
23 t
0 i
2
3
4
5
6
0
6
24
60 r(t)
-6 o
0
We have 7 points so n 6. Coefficients are easily found to be a0
8.3
=
4
8.4= a5 = a6 =
0. a2
-
Substituting in (44) we find that the fitted curve will be f(x) +_3(6_2x)
+(15_12x+2x2)(15_23x+9x2_x3).
= X3 - 6x + l].x - 6
.

23 A
BIBLIOGRAPHY
Fisher, Arrie.
Mathenatical Theory of Probability.
The MacMillan Co., 1930. Second edition.
New York,
Jackson, Dunham. Aierion Mathematical Society Colloquium Pub- lications. Vol. Xl., 1929.
Jordan, Charles. Calculus of Finite Differences. Budapest,
Rottig and Romwalter, 1939. First edition.
Mi1ne-Thoipson, L. M. The Calculus of Finite Differences.
Greenwich, England, MacMillan and Co., ]1933. First edition.
Rider, Paul R. An Introduction to Modern Statistical Methods.
St. Louis, Mo., John Wiley and Sons, 1939. First edition.
Steffensen, J. F. Interpolation. Copenhagen, Deninrk, Williams and Wilkins Co., 1927.

24
Table A
These tables are constructed by the application of (25). Lack of space has made it necessary to omit the right half of those for large n, but the reader will have no difficulty in obtaining the completed forrrt if he takes advantage of the synmietry aboixt a vertical line at the middle. That is, for t an even number the right half of any row is identical with the left half; for t an odd ntmiler the right half is numerically identical bixt each cell has an opposite sign to that of its oorrespondin cell on the left.
Although these tables are useful for curve-fitting their chief value lies in their fundamental nature. It has been seen that they were used to construct the type B tables; it is quite possible that they will be useful in the study of other interesting but as yet im- known functions.
Table B
Type B tables are used exclusively for the calculation of skew frequency distributions. If we are to consider a distribution of n + 1 points we turnì to that table headed by n n, p = 3, 4, 5, etc., depending on the degree of approximation desired. Picking an ai-bi- trary x we sum the products of f(s0)P(x,s0), f(sl)P(x,sl) etc.
Dividing this sum by 2" will give us f(x).

25
Table C
These so-called tables are merely lists of the Hermitian polynoni!- als in the finIte form. They are used in curve-fitting by polynomials as was demonstrated in Chapter 1V.

Table A n-6
3 2
1
2
3
4 t; s o
5
6
0 1
4 5 6
1
1
6
4
1 2
0 1
1
-2
1 -4
1 -6
15
5
-1
-3
-1
5
15
20
0
-4
0
4
0
20
15 6 1
-5 -4 -1
-1
3
2 1
0 -1
-1 -2 1
-5 4 -1
15 -6
1
2
11=7
3 4
1
2
3
4
5
C
7 s t o
0 1 5 6 7
1
1
1
5
3
1 1
1 -1
1
-3
1
-5
1 -7
7.
21 35
9
1
-3
-3
1
9 -5
21 -35
5
-5
-3
3
5
35
-5
-5
21 7 1
-9 -5 -1
3
3
-5
1
-1
3
3
1
3 -1 -1
-3 -1
1
-1
-5 9
35
-2].
-5 1
7 -1

Table A
2 n=
8
3 4 s t
0
1
2
3
4
5
6
7
8
0 1 5 6 7 8
1
1
1
1
8
1 0
1 -2
1
-4
1 -6
1
-8
6
4
2
28
14
4
-2
-4
56
14
-4
70
0 -14 -14 -6 -1
-10
56
-4
28
4
8
4
1
1
-6
0
0
6
6
0
2 -2 -1
-4 0 1
-2 6 0 -6 2 2 -1
4 4 -10
14 -14
28 -56
4 4 -4 1
0
70
14
-56
-14 6 -1
28 -8 1
27 n=9
4
3
1
2
3
4
5
6 s t
0
7
8
9
0 1 2
5 6 7 8 9
1 9
7 1
1
1
5
3
1 1
1 -1
1 -3
1
-5
1 -7
1 -9
36
20
8
0
-4
-4
0
84 126 126 84 36 9 1
28 14 -14 -28 -20 -7 -1
0 -14 -14 0 5 5 1
-8
-4
-6
6
6
6
8 0 -3 -1
-4 1 1
4 6 -6
-4
-4 4 1 -1
8 -6 -6 8 0 -3 1
8 0 -14 14 0 -8 5 -1
20
36
-28
-84
14
126
14
-126
-28 20 -7
84 -36 9
1
-1

28
8 9
10
0 45 10 1
18
-27 -8 -1
8 13 6 1
8 -3 -4 -1
'8
-3
2 1
0 5 0 -1
8 -3 -2 1
8 -3 4 -1
.8
13 -6 1
8 -27 8 -1
:0
45 -10 1

Table A
29
4 n - 11
5 6
2
3
4
5 s t
0
1
6
7
8
9
10
11
0 1 2 3 7 8 9 10 11
1
1
1
11 55
9 35
7 19
165 330 462 462 330 165 55
75 90
11 1
42 -42 -90 -75 -35 -9 -1
2].
-6 -42 -42 -6 21 19 7 1
1
1
1
5 7
3 -1
1 -5
-5 -22 -14
-11 -6
-5 10
14 22 5 -7
14 14 -6 -11 -1
10 -10 -10 5 5
-5
3
-1
1
1
1
1
-1 -5
-3 -1
-5
'T
-7 19
5 10 -10 -10 10 5 -5
11 -6 -14
5 -22 14
14
14
6
-22
-11
5
1
7
-21 -6 42 -42 6 21 -19
-75 90 -42
-1
3
-5
7
-42 90 -75 -35 -9 1 -9 35
1 -11 55 -165 330 -462 462 -330 165 -55 11
-1
1
-1
1
-1
1
-1
1
-1

Table A
6
7
8
9
10
11
12
13
3
4
5 s t
0
1
2 n = 13
0 1 2 3
1
1
1
3.
3.
13
11
9
7
5
3
1
-1
-3
-5
-7
-9
-11
-13
1
1
1
1
1
1
1
1
1
6
14
10
-14
-66
-154
-286
286
154
66
14
-10
-14
-6 -6
-6
-2
6
18
34
54
78
78
54
34
18
6
-2
4 5 6
15
15
-5
-29
-25
55
275
715
715
275
55
-25
-29
-5
-3.5
-25
9
63
33
-297
-1287
1287
297
-33
-63
-9
25
15
1716
132
-132
-36
36
20
-20
-20
20
36
-36
-132
132
1716
31

E
32 n = 14 s t
0
0 1
1
1
1
1
1
1
1
1
1
-6
-8
-10
-2
4
2
0
-2
-4
14
12
10
8
6
2 3 4 5 6 7
1
-5
-7
-5
91
65
43
25
11
1
11
25
43
5
91
4
-32
-100
-208
-354
364
208
100
32
-4
-16
-12
0
12
16
100].
429
121
-11
2002
572
22
3003
429
-165
-99
3432
0
-264
-88
0
-39
-19
9
-38
20
30
27
45
-5
-35
72
0
-40
2]. 0
-30 9
-19 -20
-39 38
-11 88
121 -22
429 -572
1001 -2002
-5
45
0
40
0
27 -72
-99
-165
429
0
264
0
3003 -3432

Table A
33
4
5
6
7
8
9
10
11
12
13
14
15
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 3
1
-1
-3
-5
-7
-9
-11
-13
-15
15
13
11
9
7
5 n = 15
2 3 4 5 6 7
5
-3
-7
-7
-3
105
77
53
33
17
5
17
33
53
77
105
15
-7
-57
-143
-273
-455
-17
-7
7
17
455
273
143
57
7
-15
1365
637
221
21
-43
-35
3003
1001
143
-99
-77
1
-3
21
21
39
21
-21
-39 -3
-35
-43
21
-1
77
99
.221 -143
637 -1001
1365 -3003
5005
1001
-143
-187
-11
65
6435
429
-429
25
-35
-35
25
-35
35
65
45
-45
-99 -11
-107
-143
1001
99
429
-429
5005 -6435
-99
99
45
-45

Table A
34 n = 16
'
5
6
7
3
9 io li
12
13
14
15
16
) i
4
3 o i
8
6
4
16
14
12
10
2
0
-2
-
-6
-8
-10
-12
-14
-16
1
1
1 i
1
1 i
1 i
1
1
1
1 i i
1
1
2 3 4
5 6 7 8
120
90
64
64
90
120
42
24
10
0
-6
-8
-6
0
10
24
42
10
-24
-90
-196
-350
-560
560
350
196
90
24
-10
-20
-14
0
14
20
1820
910
364
78
-36
-50
-20
14
-78
-120
-34
36
28
14
0
-42
-20 -36
-50 34
-36 120
78 78
364 -364
910 -1638
1820 -4368
4368
1638
364
42
8008 11440 12870
2002 1430 0
0
-286
-88
66
-572 -858
-286
88
0
198
110
-20
0
-90 64
-14 -70 0
-56
-14
64
0
70
20
70
0
-90
66
-88
-286
-110
-88
286
0
198
0
0 572 -858
2002 -1430 0
8008 -11440 12870

Table A 35 n 17
2
3
4
5
6 s t o i
7
8
15
16
17
12
13
14
9
10
11
O i
1
1
1 i
1
1
1
1
1 1
-1 1
1
1
1
1
-3
-5
-7
-9
1
-11
1
-13
1 -15
1 -17
9
7
5
3
17
15
13
11
2 3 4 5 6 7 8
136
104
76
52
32
16
4
0
-20
-4
-8
-8
-20
-8
-4
4
8
20
20
16
32
0
-48
52 -132
76 -260
104 -440
136 -680
680
440
260
132
48
2380
1260
560
168
-12
-60
-40
6188
2548
728
0
-156
-84
16
0
28
56
28
-28 28
0
-40
-60
-12
168
-56
-16
84
156
0
560 -728
1260 -2548
2380 -6188
12376
3640
364
-364
-208
19448 24310
3432 1430
-572 -1430
-572 -286
0 286
32
100
28
-56
176
44
-84
-56
110
-110
-70
70
-56
28
56 70
-70 84
100 -44
-176
-110
110 32
-208
-364
0
572
286
-286
364
3640
572 -1430
-3432 1430
12376 -19448 24310

36
Table A n = 18
15
16
17
18
9
10
11
12
13
6
7
8
3
4
5 s t
O o i i
18 i
2 i 16
1
14
1 12
1
10
1 8
1
1
1
6
4
2
1 0
1
-2
1
-4
1
-6
1 -8
1
-10
1 -12
1 -14
1
-16
1 -18
2 3 4 5 6 7 8 9
153 816 3060 8568 18564 31824 43758 48620
119 544 1700
89 336 820
3808
1288
6188
1092
7072 4862 0
-208 -2002 -2860
63 184
41 80
300
36
-60
168
-168
-144
-364
-364
-52
-936
-208
-858
286
0
572
23 16 208 286 0
9 -16
-1 -24
-50
-20
-24
56
116
84
144
-56
-66 -220
-154 0
-7 -16 20 56 -28 -112 14 140
-9 0 36 0 -84 0 126 0
-7 16 20 -56 -28 112 14 -140
-1 24 -20 -56 84 56 -154 0
9 16 -60 24 116 -144 -66 220
23 -16
41 -80
-60 144 -52 -208 286 0
36 168 -364 208 286 -572
63 -184
89 -336
300 -168
820 -1288
119 -544 1700 -3808
-364
1092
936 -858 0
208 -2002 2860
6188 -7072 4862 0
153 -816 3060 -8568 18564 -31824 43758 -48620

37
Table A n 19
8
9
10
11
12
13
14
15 t s
O o i i i 2 3
4
17 135 663 2244
5 6 i 19 171 969 3876 11628 27132
5508 9996
2
2380
3
1 15 103 425 1156
1
13 75 247 484
2108
468 -196
4
5
1
11
1 9
51 121
3].
39
116
-44
-132
-204
-532
-196
6 1 7
92
7 1 5
15 -7
3 -25
-76
-44
-84
36 140
16
17
18
19
7
50388
13260
8
75582
11934
9
92378
4862
884 -2210 -4862
-1300 -1794 -858
-572 858
156
78
494 286
260
28
78
-210
-286
-154
1 3 -5 -23 4 76 28 -140
-84
-98 154
1 1
-9 -9 36 36
-36
-84 126 126
1 -1 -9 9 36 -84 84 126 -126
1
-3 -5 23 4
-44
-76 28 140 -08 -154
1
-5 3 25 -36 140 -28 -210 154
1 -7
1
-9
1 -11
15 7
31 -39
51 -121
-76
-44
116
84
204
132
1 -13 75 -247 484 -468
1
-15 103 -425 1156 -2108
92
-196
-532
-196
-260
-156
572
78
494
78
1300 -1794
-884 -2210
286
-266
-858
858
4862 2380
1 -17 135 -663 2244 -5508 9996 -13260 11934 -4862
1
-19 171 -969 3876 -11628 27132 -50388 75582 -92378

Table A
12
13
14
15
16
17
18
19
20
E
10
11 t s
0
1
2
4
5
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
8
6
4
14
12
10
20
18
16
2
-8
-10
-12
-14
-16
-18
-20
0
-2
-4
-6 n = 20
2 3 4
5
-10
-8
-2
8
-2
-8
88
118
152
8
22
40
62
190
190
152
118
88
62
40
22
-8
-70
-172
-322
-528
-798
-1140
0
18
28
22
8
-22
-28
-18
1140
798
528
322
172
70
-160
-8
80
72
0
-72
-80
15504
7752
3264
952
-16
-248
8
160
248
16
-952
-3264
-7752
-15504
4845
2907
1581
731
237
-5
-83
-69
-19
27
-5
237
731
1581
2907
45
27
-19
-69
-83
4845
38

s t o
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Table A 39 n 20
G
7 8 9
10 11
8
176
104
-48
-120
-48
104
176
8
-400
-664
272
4488
15504
38760
38760
15504
4488
272
-664
-400
125970
25194
-1326
-3094
-494
650
338
-182
-238
42
210
42
-236
-182
338
650
-494
-3094
-1326
25194
125970
352
168
-112
-168
0
168
112
-168
-352
40
1104
1496
-3264
-23256
-77520
77520
23256
3264
-1496
-1104
-40
167960 184756
16796 0
-7072
-2652
-9724
0
167960
-16796
-7072
2652
936
780
-208
-364
56
252
1716
0
-572
0
308
936
-780
-208
364
56
-252
0
-252
-56
0
-252
0
308
0
252
-56
364
208
-780
-936
2652
0
-572
0
1716
-364
208
780
-936
-2652
7072
0
-9724 7072
-16796 0 16796
-167960 184756 -167960

Table B
X s o
1
2
3
4
5
6
0
3.
-1
-3
-5
7
5
3
6
X s o
1
2
3
4
5
6
0
22
10
2
-2
-2
2
10
6 n=6 p-
=
3.
1 2
1
60
32
12
0
-4
0
12
-2
-10
-18
30
22
14
6
45
35
25
15
5
-5
-15
5 n
=6 p=2
4
2
30
30
26
18
6
-10
-30
5
26 - 64
4
3
-40 o
24
32
24
0
-40
3
6 s
4
3
2
1
0 s
X
3
20
20
20
20
20
20
20
3
6
5
4
3
2
1
0 s
X

4
5
6 s x
0
1
0
57
5
-3
1
1
-3
5
6
5
6
0
42
L
-.
-2
2
2
-10
6
12
0
-4
0
12
1 n=6 p=3
2
60 -30
30
38
18
-6
-10
30
3 5 n=6 p
=4
4
1 2 3
30
42
14
-6
-2
10
18
5 4
Z-64
-45
35
39
15
-5
-5
15
20
-20
20
44
20
-20
20
3
3
-40
0
24
32
24
0
-40
1 o s
X
6
5
4
3
2
6
5
4
3
2
1 o s
X
41

2 - 128

7 6 5
2 a 128
4
7
6
5
4
3
2 i o s
X
7
6
5
4
3
2 i o s
X
43

Table B
1 n=7 p
=4
2
1
2
3
4
5
6
7 s x
0
0
-1
3
-1
-5
99
15
-5
15
7
6
7
3
4
5
1
2 s x
0
0
1
0
-2
4
-6
120
6
-4
2
42
96
22
-12
2
8
-18
28 n 7 pn5
2
3
-84
66
80
30
-12
-6
24
-42
70
-60
50
88
30
-20
10
0
7
105
69
25
-3
-7
5
9
-35
6
-105
75
79
27
-9
-5
15
-21
5
4
3
7
6
5
4
3
2 i o s
X
-35
-15
45
-73
45
-15
-35
105
4
3
2
1 o
7
6
5
4 s
X
6
5
2 128
44

Table B o
9
7
5
3 i
-1
-3
-5
-7 i
2
3
4
5
6
7
8 s
X
0
4
5
6 i
2
3
7
8 s
X
0 37
21
9 i
-3
-3 i
9
21 i
56
44
32
20
8
-4
-16
-28
-40 n=8 p=1
2
140
112
84
56
28 o
-28
-56
-84
7 i
168
100
48
12
-8
-12 o
28
72
5 6 n=8 p
=2
2 3
252
168
100
48
12
-8
-12 o
28
36
-16
-84
-168
56
84
96
92
72
6 5
3
56
28 o
-28
-56
168
140
112
84
-210
-70
30
90
110
90
30
-70
-210 rl
8
7
6
5
4
3
2 i o s
X
8
7
6
5
4
3
2 i o s
X
4
70
70
70
70
70
70
70
70
70
45

Table B
2
3
4
5
6
7
8
3
4
5
6
7
8 s
X o i o
5
-5
-35
-5
-o
3
93
35
5
8
X s o o
3
-5
-5
35
163
35
-5
-5
3
8 i
280
128
40 o
-8 o
8 o
-40
7 6 n=
8 p
=4
2 i
280
128
40 o
-8 o
8 o
-40
-3.40
140
148
60
-12
-20
20
28
-140
-280 o
120
128
72 o
-40 o
168
140
140
108
60
12
-20
-20
28
140
11=8 p=3
2 3
-280 o
120
128
72 o
-40 o
168
6
'4 rd
-210
-70
30
90
110
90
30
-70
-210 r
8
7
6
5
4
3
2 i o s
X
4
210
-70
-30
90
146
90
-30
-70
210
'41
8
7
6
5
4
3
2 i o s
X

Table B
6
7
8 s x
0
1
2
3
4
5
0
219
21
-9
1
3
-3
-1
9
-21
8
1
168
156
48
-12
-8
12
0
-28
72
7 n=8 p=5
2 3
-252
168
156
48
-12
-8
12
0
-28
56
-84
96
164
72
-36
-16
84
-168
6
28
- 256
5
4
210
-70
-30
90
146
90
-30
-70
210
4
2
1
0 s x
8
7
6
5
4
3
47

Table B
O
46
28
14
4
-2
-4
-2
4
14
28
9
6
7
8
9
1
2
3
4
5 s x
0
].
252
158
84
30
-4
-18
-12
14
60
126
=9
2
1D
2 3
504
336
200
96
24
-16
-24
0
56
144
0
-56
-112
-168
336
280
224
168
112
56
8 7 6
4
-252
-56
84
168
196
168
84
-56
-252
-504
5
9
8
7
6
5
4
3
2
1
0 s x
2 - 532
49

Table B
1
2
3
4 s x
0
5
6
7
8
9
0
130
56
14
-4
-6
0
6
4
-14
-56
9
1
-16
-6
12
14
-24
-126
504
242
84
6 n=9 p
=3
2
24
-16
-24
0
56
144
504
336
200
96
3
-336
56
224
232
144
24
-64
-56
112
504
8 7 6
4
-756
-224
84
216
220
144
36
-56
-84
0
5
9
8
7
6
5
4
3
2
1
0 s x
2 -512
50

51

Table B
0
382
56
-14
-4
6
0
-6
4
14
-56
9
2
3
4 s x
0
1
5
6
7
8
9
1
504
270
84
-6
-16
6
12
-14
-24
126
8
-504
336
312
96
-24
-16
24
0
-56
144 n=9
=5 p
2 3
-336
-56
224
280
144
-24
-64
56
112
-504
7
6
4
756
-224
-84
216
292
144
-36
-56
84
0
5
2 - 512
9
8
7
6
2
1
0
5
4
3
52

Table B
0
11
9
7
5
3
1
-1
-3
-5
-7
-9
10
2
6
7
8
3
4
5 s
X
0
1
9
10
1
-22
-38
-54
-70
90
74
58
42
26
10
-6
9
2 n = 10 p=l
3
315
45
-9
-63
-117
-171
-22
26].
207
153
99
600
504
408
312
216
120
24
-72
-168
-264
-360
8 7
210
1024
4
42
-42
-126
-210
630
546
462
378
294
210
126
6
5
5
10
9
8
7
6
5
4
3
2
1
0 s
X
252
252
252
252
252
252
252
252
252
252
252
53

Table B o o
-4
-4
0
8
20
36
56
36
20
8
10
7
8
9 io
3
4
5
6 i
2
X s o
1
8
-20
-24
-4
40
108
200
360
236
136
60
9 n = 10 p=2
2 3
4
900
612
376
192
60
-20
-48
-24
52
180
360
8 7
210
- 1024
0
-48
-64
-48
0
960
720
512
336
192
80
0
-224
-504
-840
0
168
280
336
336
280
168
6
5
-1008 10
-504 9
-112
168
336
392
8
7
6
5
336
168
-112
-504
-1008
4
3
2
1
0
5 s x

Table B
6
7
8
9
10
2
3
4
5
X s o i o
4
8
0
-28
-84
176
84
28 o
-8
-4
10
1
8
28
8
-84
-280
840
428
168
28
-24
-20
9 n = 10 p=3
2 3 4 5
1260
756
400
168
36
-20
-24
0
28
36
0
0
80
-64
-112
0
336
960
0
336
448
400
256
7
-1680
-504
168
448
448
280
56
-112
-112
168
840
-1008
-504
-112
168
336
392
336
168
-112
-504
-1008
6 5
10
9
8
7
6
5
4
3
2
1
0 s x
10
2 1024
55

Table B 56 o
386
126
14
-14
-6
6
6
-6
-14
14
12C
6
7
S
9
10
3
4
5 s
X o i
2
10
1
1260
512
140
0
-20
0
12
0
-20
0
140
9 n = 10
=4 p
2 3
4
630
630
538
210
30
-50
-30
42
70
-90
-630
-1680
0
560
512
240
0
-80
0
112
0
-720
-1260
-420
140
420
452
300
60
-140
-140
252
1260
8
7
210
- 1024
6
5
5
6
5
4
3
2
1
0 s x
10
9
8
7
1512
0
-280
0
360
512
360
0
-280
0
1512

Table B 57
6
7
8
9
10
3
4
5 s x
0
1
2
0
6
-6
-6
14
14
-126
638
126
-14
-14
6
10
1
1260
512
140
0
-20
0
12
0
-20
0
140
9 n = 10
=5 p
2 3
4
-630
630
582
210
-30
-50
30
42
-70
-90
630
-1680
0
560
512
240
0
-80
0
112
0
-720
1260
-420
-140
420
572
300
-60
-140
140
252
-1260
6 8 7
10
2 =1024
5
5
4
3
2
1
0
10
9
8
7
6
1512
0
-280
0
360
512
360
0
-280
0
1512
5 x

Table B 58
1
2
3
4
5
6
7
8
9
10
11 x s
0
0
12
10
4
2
8
6
0
-2
-4
-6
-8
-10
11
1
-16
-34
-52
-70
-88
110
92
74
56
38
20
2
10
2
440
370
300
230
160
90
20
-50
-120
-190
-260
-330 n = 11 p=1
3
990
840
690
540
390
240
90
-60
-210
-360
-510
-660
9 8
2h
- 2048
4
1320
1140
960
780
600
420
240
60
-120
-300
-480
-660
7
5
6
11
10
5
'1
3
9
8
7
6
2
1
0 s x
252
168
84
0
588
504
'120
336
924
840
756
672

Tahi B 59
7
8
9
10
11
1
2
3 s
X o u
0
67
45
27
13
3
-3
-5
-3
3
13
27
45
11
1
3].
-15
-33
-23
15
'.95
337
207
105
8].
175
297
10
1485
1036
66].
363
141
-5
-75
-69
13
171
405
715
2 n = 11 p=2
3
-15
-81
-63
39
225
495
2145
1575
1089
687
369
135
9 8
211
. 2048
4
5
990
930
846
738
606
450
270
66
-162
414
-690
-990
-1386
-630
-42
378
630
714
630
378
-42
630
-1386
-2310
7 6
11
10
9
8
7
6
5
4
3
2
1
0 s x

Table B 60 x s
0
0
8
-8
-48
-120
232
120
48
8
-8
-8
0
8
1
2
3
4
5
6
7
8
9
10
11
11 n = 11 p=3
1 2 3 4
5
1320
712
312
80
-24
-40
-8
3 2
40
-24
-200
-528
2640
1560
808
328
64
-40
-40
8
48
24
-120
-440
1320
1200
984
712
424
160
-40
-136
-88
144
600
1320
-2640
-720
384
848
348
560
160
-176
-272
48
960
2640
-3696
-1680
-336
448
784
784
560
224
-112
-336
-336
0
10
7 6
9 8
211
- 2048
11
10
9
8
7
6
5
4
3
2
1
0 s x

61
Table B
5
6
3
4
7
8
9
10
11 s
X o i
2
0
10
2
-14
-14
42
210
562
210
42
-14
-14
2
11
1
2310
982
294
14
-42
-10
22
14
-26
-42
70
462
10 n = il
=4 p
2 3 4 5
2310
1470
814
350
70
-50
-50
14
70
30
-210
-770
-2310
210
1050
954
490
50
-150
-70
154
210
-390
-2310
-4620
-1260
420
980
884
500
100
-140
-140
84
420
660
924
-420
-420
140
700
924
700
140
-420
-420
924
4620
9 8 7 6
2h
2048
7
6
5
4
3
2
11
10
9
8
1
0 s x

Table B
62
1
2
3
4 x s
O
O
7
O
-12
0
9 28 io o
II
-252
1024
252
0
-28
0
12
11
1
0
12
0
-12
0
28
0
2772
1024
252
0
-28
10 n =
II p=5
2 3 4
O
84
0
-180
0
1540
0
1260
1024
420
O
-100
-4620
0
1260
1024
420
0
-100
0
84
0
-180
0
O
-840
O
840
1024
600
0
-280
0
504
0
-3960
9 8 7
5
0
1
0
5
4
3
2 s x
11
10
9
8
7
6
0
-280
0
504
0
5544
O
-840
O
840
1024
600
1].
2
2048

Table B
5
6
7
8
9
10
11
7
-7
-9
-11
-1
-3
-5
5
3
1
12 x s
0
13
11
1 2 n 12 p=1
3
4
5
132
112
02
72
52
32
12
-8
-28
-48
-68
-88
-108
-22
-110
-198
-286
-374
-462
594
506
418
330
242
154
66
1540
1320
1100
880
660
440
2475
2145
1815
1485
1155
825
495 220
0
-220
-440
-660
-880
-1100
165
-165
-495
-825
-1155
-1485
792
528
264
0
-264
-528
-792
2376
2112
1848
1584
1320
1056
11 10 9 8 7
212
- 4096
6
6
12
11
10
9
8
7
6
5
4
3
2
1
0 s x
924
924
924
924
924
924
924
924
924
924
924
924
924

Table B
64
9
10
7
8
1J.
12
5
6
0
1
2
3
4 7
-1
-5
79
55
35
19
19
35
55
-5
-1
7 x 0
12 n 12 p2
3 4 1 2
5 6
660
464
300
168
68
0
-36
-40
-12
48
1'O
264
420
-126
-58
114
390
770
1254
2310
1650
1094
642
294
50
-90
11 10
-160
-140
40
380
880
1540
4180
3080
2140
1360
740
280
-20
9
105
-135
-315
-435
-495
-495
3465
2805
2205
1665
1185
765
405
720
168
-576
-1512
-2640
-3960
-792
0
600
1008
1224
1248
1080 1428
1260
756
-84
-1260
-2772
-4620
-4620
-2772
-1260
-84
756
1260
8 7
6
5
4
3
2
1
0 s x
8
7
6
12
11
10
9
212
= 4096

able B 65 n = 12 p
=3
9
10
11
C
7
8
2
3
4
5
X s o
-i
5
11
5
-2].
-75
-165
299
165
75
21
-5
-11
-5
1 2 3
20
60
36
-100
-396
-900
1980
1124
540
180
-4
-60
4950
2970
1574
666
150
-70
-90
-6
86
90
-90
-550
-1386
-20
-140
-116
36
300
660
1100
4620
3300
2220
1364
716
260
4 5 6
405
-165
-459
-261
645
2475
5445
-2475
-165
1125
1611
1509
1035
-8712
-3960
-840
936
1656
1608
1080
360
-264
-504
-72
1320
3960
-4620
-2772
-1260
-84
756
1260
1328
1260
756
-84
-1260
-2772
-4620
12 11 10 9 8 7 6
3
2
1 o
6
5
4
9
8
7
12
11
10 s
- 4098

66 n = 12 p
=4
1
2 x s
0
6
7
8
9
10
11
12
3
4
5
0
794
330
90
-6
-22
-6
10
10
-6
-22
-6
90
330
1 2 3 4 5 6
3960
1784
600
72
-72
-40
24
40
-8
-72
-40
264
1080
5940
3300
1604
612
116
-60
-60
4
52
36
-60
-220
-396
-1320 -10890
1320 -2970
2040 870
1688
920
2070
1798
950 200
-200
-200
150
-250
-170 88
360
120
198
-1320
-4840
390
-330
-2970
-4752
-2640
-720
720
1520
1648
1200
400
-400
-720
48
2640
7920
9240
1848
-840
-840
280
1400
1848
1400
280
-840
-840
1848
9240
12 11 10 9 8 7 6
5
4
3
7
6
9
8
12
11
10
2
1
0 s x

Table B 67
= 12 p=5
3
4
5
6
1
2
9
10
11
12 x s
0
7
8
0
-10
-14
14
42
-42
-462
1586
462
42
-42
-14
14
10
1 2 3 4
5
5544
2048
504
0
-56
0
24
0
-24
0
56
0
-504
84
84
-108
-252
308
2772
2772
2772
1796
756
84
-140
-60
-9240
0
2520
2048
840
0
-200
0
168
0
-360
0
3080
150
-350
-210
378
630
-990
-6930
-6930
-2310
630
1890
1838
1050
11088
0
1008
0
-7920
0
-1680
0
1680
2048
1200
0
-560
6
9240
1848
-840
-840
280
1400
1848
1400
280
-840
-840
1848
9240
12 11 10 9 8 7 6
1
0 s x
8
7
6
5
4
3
2
12
11
10
9
212
- 4096

Table B 68
0
5
6
3
4
7
8
9
10
11
12
13 s
0
1
2
-6
-8
-10
-12
0
-2
-4
6
4
2
14
12
10
8
13 n = 13 p=1
3 4
1
2
-20
-42
-64
-86
-108
-130
156
134
112
90
68
46
24
24
-84
-192
-300
-408
-516
-624
780
672
564
456
348
240
132
132
-176
-484
-792
-1100
-1408
-1716
2288
1980
1672
1364
1056
748
440
-110
-660
-1210
-1760
-2310
-2860
4290
3740
3190
2640
2090
1540
990
440 990
396
-198
-792
-1386
-1980
-2574
5148
4554
3960
3366
2772
2178
1584
3432
3168
2904
2640
2376
2112
1848
1584
1320
1056
792
528
264
0
12
2
11 10 9
5
8
6
7
9
8
7
6
13
12
11
10
5
4
3
2
1
0 s x
213 -
8192

Table B 69 n = 13 p=2
9
10
11
12
13 s x
0
1
2
3
4
5
6
7
8
0
26
12
2
-4
92
66
44
-6
-4
2
12
26
44
66
1 2 3 4 5 6
3432
2508
1720
1068
552
172
-72
-180
-152
12
312
748
1320
2028
858
620
418
252
122
23
-30
-52
-38
12
98
220
378
572
7436
5544
3916
2552
1452
616
44
-264
-308
-88
396
1144
2156
3432
8580
6710
5060
3630
2420
1430
660
110
-220
-330
-220
110
660
1430
2574
2772
2838
2772
2574
2244
1782
1188
2676
2412
462
-396
1620
300
-1584 -1386
-2508 -3960
-3762 -6864
-5148 -10296
-6864
-3960
-1584
300
1620
2412
13 12
1].
10 9 8 7
8
7
6
13
12
11
10
9
5
4
3
2
1
0 s x
213
= 8192

n = 13 p
=3
7
8
9
10
11
12
13
: s o
1
4
O i 2 3 4
5 6
-2
-40
-110
-220
0
10
12
378
220
110
40
2
-12
10
2860
1698
880
350
52
-70
-72
-10
60
82
0
-242
-700
-1430
8580
5280
2908
1320
372
-80
-180
-72
100
192
60
-440
-1452
-3120
-180
-112
52
200
220
0
-572
11440
7700
4840
2748
1312
420
-40
1430 -15444 -17160
2860 -6930 -5O4
3410 -1320 -3960
3280 1890 -240
2670 3204 1944
1780 2880
810
3126
2160 2856
-40 2160
-570
810
-420
-580 -1026
1080
-96
130 -1080 -504
1650 -1584 1760
4510
8580
5940
12870
-1320
0
13 12 11 10
9 6 7
1
0 s x
8
7
6
5
4
3
2
13
12
11
10
9
213
8192

Table B n = 13 p
=4
3
9
10 ii
12
13
. i
2
X s o
4
5
7
O
1093
495
165
15
-27
-17
5
15
5
-17
-27
15
165
495
1 2
3
4
5 6
6435
3073
1155
225
-93
-95
3
65
35
-63
-125
33
675
2145
12870
6930
3238
1170
198
-110
-90
18
70
18
-90
-110
198
1170
4290 -19305 -21879
4950 -5115 -9405
4290 1815
2998 4005
-1815
2115
1602
470
-190
-330
-62
3511
1925
375
-475
-425
3465
3171
2025
675
-375
342
450
261
855
-330
-2750
165
-3465
-7722 -12155
-765
-279
1155
3465
6435
8580
396
900
-1140
-1980
396
8580
25740
-1980
-1140
900
2700
3396
2700
13 12 11 10
9 8
't
12
11
9
8
7
6
5
4
3
2
1
0 s x
213 - 8192

Table B 72 n = 13 p
=5
5
6
7
8
9
10
11
12
13
2
3
4
: s
0
1
0 1 2 3
4
5 6
132
-48
-36
8
20
0
-20
-8
36
48
-132
-792
2380
792
10296
3964
1056
36
-120
-20
48
20
-40
-36
64
132
-216
-1716
10296 -13728 -25740
6336 792 -6600
3304
1296
216
4752
3880
1728
1980
4320
-160
-120
48
120
0
-216
-176
792
3744
120
-400
-120
288
216
-432
-792
1408
10296
10296
-1980
-2640
540
3556
1800
300
-400
3240
3796
2400
300
-1000 -300
216
540
-540
1296
0 1980
-3960 -1980
-5720 -25740
34320
6336
-2640
-2400
720
3200
3696
2400
400
-960
-720
1056
2640
Q
13 12 ii 10 9 8 7
6
5
4
3
2 i o s
9
8
7
13
12
11
10
215 - 8192

Table B 73 n 14 p=1
X s o
O
1
2
3
4
5
6
7
5
8
9
10
11 -7
12 -9
13 -11
14 -13
3
1
-1
-3
-5
15
13
11
9
7
1
-10
-34
-58
-82
-106
-130
-154
86
62
38
14
182
158
134
110
2 3 4 5 6 7
1001
871
3276
2860
7007
6140
10010 9009
8866 8151
741
611
481
351
221
91
2444
2028
1612
1196
780
364
5291
4433
3575
2717
1859
1001
-39 -52
-468
143
-715 -169
-299 -884 -1573
-429 -1300 -2431
-559 -1716 -3289
-689 -2132 -4147
-819 -2548 -5005
7722 7293
6578 6435
5434 5577
4290 4719
3146 3861
2002 3003
858 2145
-286 1287
-1430 429
-2574 -429
-3718 -1287
-4862 -2145
-6006 -3003
3432
3432
3432
3432
3432
3432
3432
3432
3432
3432
3432
3432
3432
3432
3432
14 13 12 11 10 9 8 7
9
8
7
14
13
12
11
10
3
2
1
0
6
5
4 s x
214
16384

Table B
74 n = 14 p=2
6
7
8
9
10
11
12
13
14
1
2
3
4
5
8
X o
O
1 2 3 4 5 6 7
106 1092
78
54
34
18
6
-2
-6
-6
-2
6
18
34
54
78
808
564
360
196
72
-12
-56
-60
-24
52
168
324
520
756
4914 12376 18018 12012 -6006 -20592
3666
2590
9360 14014 10296 -2574 -13728
6744 10494 8668 198 -7920
1686 4528 7458 7128 2310 -3168
954 2712 4906 5676 3762 528
394 1296 2838 4312 4554 3168
6 280 1254 3036 4686 4752
-210 -336 154 1848 4158 5280
-254 -552 -462 748 2970 4752
-126 -368 -594 -264 1122 3168
174 216
646
1290
2106
1200
2584
4368
-242 -1188 -1386 528
594 -2024 -4554 -3168
1914 -2772 -8382 -7920
3718 -3432 -12870 -13728
3094 6552 6006 -4004 -18018 -20592
14 13 12 11 10 9
8 7
2
1 o
5
4
3
11
10
9
14
13
12
8
7
6
8
X
214
= 16384

Table B 75 n - 14 p-3
10
11
12
13
X
8 o i
2
3
4
5
6
7
8
9
14
O i 2 3 4 5 6 7
470 4004 14014 24024 14014 -20020 -42042 -20592
286 2472
154
66
14
-10
-14
1364
616
164
-56
-108
8866 16016 11726 -8008 -23166 -13728
5090 9944 9394 -132 -9702 -7920
2486 5552 7106 4312 -858 -3168
854 2584 4950 6028 4158 528
-6 784 3014 5720 6138 3168
-294 -104 1386 4092 5874 4752
-6 -56 -336 154 1848 4158 5280
6 36
-210
46 -168 -594 -308 1782 4752
14 104 274 144 -770 -1672 -462 3168
10 84 274 344 -286 -1540 -1782 528
-14 -88 -154 176 946 792 -1386 -3168
-66 -476 -1210 -616 3014 6028 1518 -7920
-154 -1144 -3094 -2288 6006 14872 7722 -13728
-286 -2156 -6006 -5096 10010 28028 18018 -20592
14
4
3
2
1
0
8
7
6
5
13
12
10
9
14 13 12 11 10 9 8 s x
7
214
= 16384

Table B 76 n - 14 p
=4
9
10
11
12
13
14
X s
5
6
7
8 o
1
2
3
4
O 1 2 3 4 5 6 7
1471 10010 25025 20020 -25025 -58058 -15015 51480
715
275
5046
2090
13585 14300 -5005 -24310 -11583 17160
6421 9460 4675 -4730 -6435 792
55 550 2365 5596 7535
-25 -70 425 2740 6471
4730 -1155 -3960
7510 3105 -2280
-29 -170 -215 860 3755 6442 5625 1800
-5 -54 -195 -140 1035 3750 6117 5400
15
15
-5
-29
-25
55
275
715
70
90
-10
-150
-154
250
1430
3850
21
145
65
-420
-204
220
500
-665
-945
-29
1235
1050
-650
4725 6792
5400 2025
-975 -950 1800
-58 -2835 -2280 -155
-275 220 1375 1210 -1683 -3960
121 -1100 -1705 1430 4785 792
1625 -4004 -10725 -1430 19305 17160
5005 -9100 -29029 -10010 45045 51480
14
13
12
1].
10
9
8
7
6
5
4
3
2
1
0
14 13 12 11 10 9 8 7 s x z14 - 16384

Table B 77 n 14 p
=5
X s
O
1 2 3 4 5 6 7 o
1
2
3
4
5
6
7
8
9
10
11
3473 18018 27027 -12012 -63063 -18018 75075 51480
1287 7334 14157
297 2178 6443
5148 -15873 -12870 14157 17160
9108 4257 -4290 -5445 792
198 2277 7004 9207 -33
-63
-9
25
15
-222
-90
66
70
387
-195
-165
21
3348
540
-620
-420
7193
3375
465
-665
2970 -5115 -3960
6750 1395 -2280
6842 6525 1800
4350
1050
7467
4725
5400
6792
-15 -30
-25
9
63
-90
2
198
12 33
-297
162
-856 13
14 -1287 -4158
115 276 -375 -1250 675 5400
45 540 351 -1350 -1875 1800
-117
-187
99
1053
-108
-1188
513
-297
702 -1125 -2280
2970 2277 -3960
-748 -1287
5148 143
990 3795 792
-12870 -6435 17160
3003 22932 9009 -50050 -45045 51480
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0 s
14 13 12 11 10 9 8 7
214
- 16384

78 Table B n 15 pl
4
5
6
7
8
1
2
3
12
13
14
15
9
10
11
X s o
O i
2 3 4 5 6 7
16
14
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-14
210
184
158
132
106
80
54
28
2
1260
1106
952
798
644
490
336
182
28
-126
4550 10920 18018 20020 12870
4004 9646 16016 18018 12012
8372 14014 16016 11154 3458
2912
2366
7098 12012 14014 10296
5824 10010 12012 9438
1820 4550 8008 10010 8580
1274 8008 7722 3276
2002
6006
4004 728
182 2002
6006
4004
6864
6006
-364
728
-546 0 2002 5148 -24
-50
-76
-102
-128
-280 -910 -1820 -2002 0
-434 -1456 -3094 -4004 -2002
-588 -2002 -4368 -6006 -4004
-742 -2548 -5642 -8008 -6006
-154 -896 -3094 -6916 -10010 -8008
-180 -1050 -3640 -8190 -12012 -10010
4290
3432
2574
1716
858
0
15 14 13 12 11 10 9 8
5
4
3
2
1
0
9
8
7
6
15
14
13
12
11
10 s x
2 - 32768

Table B 79 n 15 p=2
1
2
3
4
9
10
11
12
13
14
15
5
6
7
8
X s o o i 2 3 4 5 6 7 i
11
25
43
65
91
121 1365
91 1031
65
43
25 li i
-5
-7
-5
21
-49
-75
-57
741
495
293
135
5
111
261
455
693
975
1545
755
177
-189
-343
-265
-15
467
1161
2067
3185
4515
6825
5187
3761
2547
19565 34125 33033 5005 -32175
15015 26663 27027 7007 -21021
11037 20085 21593 8437 -11583
-3861 7631 14391 16731 9295
4797 9581 12441 9581 2145
2535 5655 8723 9295 6435
9009 845
-273
2613 5577 8437
455 3003 7007
-819 -819
-793 -1209
-195
100]. 5005
-429
243].
-715 -1287 -715
9867
9009
6435
2145
976 663 -1573 -4433 -3861
2717
5031
2925 -1237 -8723 -11583
6071
7917 10010
11375 15015
-429 -13585 -21021
1001 -19019 -32175
3003 -25025 -45045
15 14 13 12 11 10 9 8
8
7
6
5
4
3
2
1
0 s x
15
14
13
12
11
10
9
215
32768

[11] Table B n = 15 p=3
3
4
7
8
12
13
14
15
9 lo
11 s
X o
1
2
5
6
O 1 2 3 4 5 6 7
576
364
208 2025
100
32
-4
-16
5460
3468
-132
-12 -112
0
12
16
4
-32
-100
-364
1008
356
0
-12
96
140
48
-252
-832
-208 -1764
-3120
21840 45500 43680 -12012 -80080 -77220
14196 30576 32396 0 -44044 -48048
8480 19188 23088 7436 -18304 -25740
4428 10880 15588 11088 -1364 -9504
1776 5196 9728 11748 8272 1452
260 1680 5340 10208 12100 7920
-384 -124 2256 7260 11616 10692
-420 -672 308 3696 8316 10560
-112 -420 -672 308 3696 8316
276
480
176
660
-852 -2112 -748
-400 -2772 -3520
4752
660
236
-720
-2652
-5824
576
-532
-3120
-7644
-10500 -14560
516 -580 -3124 -3168
1728 4356 1936 -5940
3068 13728 13156 -6864
4368 28028 32032 -5148
5460 48048 60060 0
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0 s
15 14 13 12 11 10 9 8
215
32768

Table B 81 n
= 15 p
=4 i
2
9
10
11
12
14
15
X s o
3
4
5
6
7
8
13
O i 2 3 4 5 6 7
1941 15015 45045 55055 -15015 -117117 -95095 57915
1001 7947 25025 35035 5005 -49049 -51051 15015
429 3575 12237 20735 13585 -9581 -20735 -3861
121 1155 4785 11027 14685 947]. -1595 -7425
-11 55 1045 4895 11577
-39
-19
-245 -335
-153 -435
1435
-145
6845
2385
15059 8745 -2805
12903 12485 4455
7491 11649 10395
9
21
9
-19
-39
-11
121
429
1001
35 -63 -525 -595 2079 8085 12639
135 245 -273 -1575 -1309 3465 10395
75 225 155 -723 -1881 -715 4455
-105 -115
-253 -495
415
275
1105
2365
-77 -3135 -2805
2431 -2651 -7425
-105 -363 -385 825 2739 1705 -3861
715 1105 -1573 -6435 -3289 10725 15015
2695 5005 -3185 -23023 -21021 25025 57915
6435 12705 -5005 -53235 -57057 45045 135135
15 14 13 12
1].
10
9 8
5
4
3
2
1
0
8
7
6
15
14
13
12
11
10
9 s x
215
- 32768

Table B 82 n 15 p=5 s
X O
3.
2 3 4 5 6 7 o i
2
3
4
5
6
7
8
9
10
11
4944 30030 60060 10010 -120120 -114114 100100 193050 15
2002 12952 30030 20020 -30030 -48048 14014 60060 14
572 4290 12952 18590 8580
22 660 4290 12512 18150
-9438 -11440 2574 13
9372 -8030 -11880 12
-88 -330 660 6050 14272 14982 3740 -6270 11
-38 -240 1420 6810 12904 12550 4500 10
20 42
-330
-240 -730 1020 7530 14184 12150 9
30 140 42 -840 -1330 2100 9450
13584 8
0 30 140 42 -840 -1330 2100 9450 7
-30 -120 30 740 642 -1920 -3250 2700
6
-20 -110 -120 430 1140 -78 -3200 -2850
5
38 132 -110 -880 -330 2508 2354 -3960 4
12
13 -
88 390
Q
132 -1870 -2640
390 572 -1430
14 -572 -2310 0 11830 12012
15 -2002 -8580 -2310 40040 51870
2838
-3432
8140 594
1430 8580
-22022 -40040 12870
-60060 -150150
0
3
2
1
0
15 14 13 12 11 10 9 8
8 x
215
- 32768

Table C n
5
110(x) i
111(x) - 5 -
2x
H2(x) = 10 - lOx + 2x2
H3(x) 30 - 62x + 30x2
3
114(x) = 15
- 70x + 64x2 - 20x3 + 2x4
3
H5(x) = 15 - 256x + 400x2 - 220x3 + 50x4
-
15
= i
H1(x) = 6 - 2x
- 15 - 12 x + 2x2 n-6
113(x) = 60
-
92x + 36x2
3
4x3
H4(x) 45 - 132x 94x2 - 24x3 + 2x4
3
H5(x) w 90 - 606x + 720x2
- 320x3
+ 6O
15
4x5

Table C n=7
H0(x) = i
111(x) = 7 - 2x
= 21 - 14x + 2x2
H3(x) = 105 - 126x + 42x2 - 4x3
3
114(x) = 105 - 224x + 130x2 - 28x3 + 2x4
E5(x) 315 - 1266x +
1190x2
- 440x3 + 70x4 - 4x5 n=8
110(x) - i.
H1(x) 8 - 2x
112(x) - 28 - lGx + 2x2
113(x) - 168
-
170x 48x2
114(x) 210 - 688x + 172x2 - 32x3 + 2x4
H
(x) = 840 - 2386x + 1840x2 - 580x3
15
BOx4 -

Tab1 C u
9
B,(x) i
9 - 2x
I2(x) 36 - 18x + 2x2
- 252 - 218x i-
54x2 - 4x3
3
H4(x) - 378 - 522x + 220x2
- 36x3
+ 4x4
3
H5(x) 1890 - 4146x
+
2610x2
-743
+ 90x4 - 4x5
85 n 10
110(x) i
H1(x) = 10 - 2x
H2(x) - 20x + 2x2
H3(x) - 360
-
272x 60x2
114(x) 630
-
740x + 274x2 40x3
+ 2x4
II5()
- 3760 - 6756x
+
3800x2 - 92 + 100x4

Table C n 11
110(x) - i
- - 2x
H2(x) - 55 - 22x + 2x2
- 495 - 332x
+
66x2 - 4x3
990 - 1012x + 332x2
- 4-4x3 + 2x4
E5(x) - 6930
-
10456x + 5170x2 -ll20x' + llOx4 - 4x5 n 12
110(x) - i ff1(x) 12 - 2x
112(x) 66 - 24x + 2x2
H3(x) - 1320 - 788x + 132x2
3
4x3
114(x) = 1485
-
1344x + 400x2 - 48x3 +
3 ff5(x) - 11880 - 17316x
+
8640x2 - 1340x3 + 120x4 4x5