I ntrnat. J. ,lath. & Math. Sci.
Vol. 6 No.
(1983)171-180
171
A CLASS OF ORTHOGONAL POLYNOMIALS OF A NEW TYPE
M. DUTTA and
KANCHAN PRABHA MANOCHA
S.N. Bose Institute of Physical Sciences
University of Calcutta
92, Acharya Prafulla Chandra Road
Calcutta-700009, INDIA
(Received June 19, 1982)
ABSTRACT.
The purpose of this paper is to study a class of polynomials
Fm(x,%,)
n
associated with a class of m differential equations, where % is fixed, n is a variable
parameter, m is fixed positive integer and v is a non-negative integer < m.
We also
obtain all the important properties of this class of polynomials including Rodrigues’
formulas, generating function, and recurrence relations.
Several special cases of
interest are obtained from our analysis.
KEY WORDS AND PHRASES.
recurrence relion.
Function space, weight function, generating function, and
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
1
33A65, 34A30.
INTRODUCT ION
The classical orthogonal polynomials, including Jacobi, Gegenbauer, Legendre,
Tchebicheff, Laguerre and Hermite polynomials and some of their generalizations, have
been studied in great details
(Erdlyi
et al
[io]).
Each of these classes of polynom-
ials satisfies one differential equation, one Rodrigues’ formula and a pure recurrence
relation.
In 1965, Dutta and More [2] obtained the generalization of Legendre polynomials
with the help of Schmidt-orthonormalization process, in this it is seen that this
class of polynomials satisfies two different differential equations, two Rodrigues’
formulas, etc., for even and odd values of the variable parameter in the
space.
same function
172
M. DUTTA AND K. P. MANOCHA
Similar generalization of the class of Hermite polynomials, so that two differential equations, two
Rodrigues’ formulas and
two recurrence relations for even and
associated in the same function space, was
odd values of the variable parameter and
obtained, and studied in some details by Dutta, Chatterjea, Ghosh [3] and Dutta,
Chatterjea and More [4].
Earlier in the paper of Krall and Frink
[5],
some similar
result i.e. two different differential equations in the discussion of a class of functions written by suitable modification of Bessel functions are seen, but this pecular-
ity is not studied with due emphasis.
Theoretical justification of the association of a pair of differential equations
with a single class of orthogonal functions has been discussed in some details in
Dutta and Bhattacharya [6].
In a subsequent paper of Dutta [7], the possibility of a
class of orthogonal polynomials having m number of differential equations, m recurrence
relations, etc. (m being a.fixed natural number) is established.
In this paper following the suggestions of Dutta [7], it is shown how and when
solutions of m differential equations may belong to the same class of orthogonal poly-
nomials.
This new class of orthogonal polynomials satisfies m
Rodrigues’ formulas,
m
recurrence relations, etc., where m being the number of partitions of the variable
parameter n.
Let us consider a class of m differential equations:
m-i
y"
where
(m x
x
)y’ + (mn xm-2
( +
-
x
2
i)
)y
(i i)
0
is fixed, n is a variable parameter, m is a fixed positive integer where
is
a non-negative integer < m.
Here we see that for different values of v we shall obtain m different differential equations, but solutions of these differential equations are orthogonal in the
w(x,%)(i)
2
same function space L
where
w(x,%)
I
(0, )
e
-x
m
Ixl +m-2
when m is odd,
(_oo,m) when
m is even.
Of course for an odd m, as I is (0, ) so
Ix
is obviously replaced by x.
The in-
tervals are taken from the consideration of convergence of the integral defining
CLASS OF ORTHOGONAL POLYNOMIALS OF A NEW TYPE
173
or tho gonali ty.
{Fm(x,l
)}
n
THE CLASS OF POLYNOMIALS
2.
AND THEIR ORTHOGONALITY.
Solving the differential equation (I.i), we obtain a class of functions
n-u
k
n-
[--]
Fm(x,l,
)
n
+
2+ %
k=0
for running parameter n and for fixed
=
(mod m) and
for n
)k (xm)
m
x
%,
(2.1)
m- i
k’
k
m
m the functions will reduce to polynomials
0,1,...,m-l, where is a non-negative integer < m.
Now all
(mod m) form a class of orthogo-
the polynomials for different values of n with n
nal polynomials.
corresponding to a particular (fixed) value of m we
For differnet values of
shall obtain m different classes of polynomials.
Following the technique of Dutta and Bhattacharya [6] and Dutta
ORTHOGONALITY.
m
[7],
we can calculate the weight function
w(x,l)
e
-x
Ixl l+m-2
of the function space
properly associated with the class of differential equations and establishes the or-
thogonality relation:
m
e
-x
Ixlx+m-2 Fi (x,X
m
)
F,(x,X, )
j
(2.2)
6
dx
ij
where
I
(0,o)
if m is odd,
(-o%00) if m
3.
THE POLYNOMIALS AS
is even.
2F0
-
From the equation (2.1) by reversing the order of summation, we can write at once
n
n
(x,l, )
(-I)
m
2+ I + m
m
1
x
)n,,-
o
n
2F0
n
+ + I
i
m
m
x
(3.1)
m
4.
RELATION WITH LAGUERRE POLYNOMIALS.
In (2.1) replacing x by
and n by
mn+
,
we obtain
--(here
we are taking the real positive root of
/)
174
M. DUTTA AND K. P. MANOCHA
v n
A’)
k
x
(-n)k
x
+
2+ %
k=0
m- i
k.’
m
-n
IFI
x
(4.1)
x
+
2+
m- I
m
(rood m).
where n
This can be written in terms of Laguerre polynomials:
x
(N/ x’%’ )
nn+
2+
m
n!
2+ % + m
L
1
m
5.
RODRIGUES
-
I
m
n
n
(4.2)
(x)
FORMULAS.
For Laguerre polynomials, Rodrigues’ formula is given by
L
(a) (x)
e
x
n!
n
n
D
(e
-x
xn+)
d
dx
D
where
So, from (4.2), we get
m
n!
x
(r,,)
+
2+ %
m
x x
m- i
m
e
x
n
D
n!
,n
(e-X
+
2+%-I
m
x
n
or
m
Fmn+
m
x
2+ )
(r-,, v)
e
+
x
m- i
m
6.
(e -x x
n
+
m
(5 I)
n
Obviously, for different values of v (
erent
n
D
m-l), we shall obtain m diff-
0,1,2,
Rodrigues’ formulas.
GENERATING FUNCTIONS.
Equation (2.1) can be written as:
n
+)(x,l,V)
x
.
k=0
(_n)
2x)+
k
m)k
(x
I + m-
i
m
(6.1)
k!
)k
n
Multiplying both sides by
Z
n=0
TM
F
mn+"x’
)
tn
x
n=0
and taking the sum from n
n
Z
k=0
(-n) k
2+
+
m
(xm) k
0 to
tn
m- I
k
k
n!
oo,
CLASS OF ORTHOGONAL POLYNOMIALS OF A NEW TYPE
x
x
n
tn
(-i)
n=0! k=0
-
n=0
t
i
m
)k
m
k:
xmkt
k
mn+)(x’%’v) !
x
e
xmk tk
(-i)
2+l+m- i
t
k!
)k
m
Fm(x,%
n
n
k
)k
m
n
t
Thus the generating function for
m
+
k
n+k
(-i
2+%+ m- i
n=0 n! k=0
F
xmk
n!
2+ %
(n- k)!(
n!
n=0 k=0
=x
k
175
) is given by the relation
2+%+m- i
0FI(--;
x
m
m
(6.2)
t)
To find out the other generating function multiplying both sides of the equation (6.1)
n
t
by
and taking the sum from 0 to oo,
(C)n
i
n=0
(c) n Fm
mn+ v(x,%, )
tn
i
(c)
x
n=0
k
n
0
(C)n+k
x
n!
n,k=0
(I
k)n
m
-c
x
+
n
m- I
)k
xmk
t
)k
(C)k
k!
n+k
k!
(-i)
xmk
k
)
t
k
2+l+m- 1
k
k
)k
k
xmk tk
k! (i
t)
c+k
(C)k (-xm t)k --,(6.3)
2x)+l+m- 1
k
)k (i- t) k!
m
equation (6.3) can be written as:
m
)n ’mn+V(x’l’)
k
n
2v+. I + m- i
t)
2+l+m-I
_m
t
xmk
k
m
k=0
2 +l+m-i
(-i)
n
n!
k!
m
(c) k (-i)
k=0
With the choice c
(-i)
n!
n=0
x
t
k
2x)+t+m-1
m
(c +
k=0
x
m- i
2v+ X
k)!(
(n
k=0
n=0
+
m
(c)
i
X
(-n)k
2+ %
t
n
(i
t)
Hence our required generating function is given by
2 x)-t-)-t-m- 1
m
m
x
0F0(
(i
t)
M. DUTTA AND K. P. MANOCHA
176
2+ %
+
m
1
m
n=0
7.
m
mn+ (x
F
)n
% )
.
2+l+m-I
tn
x
v
m
(i- t)
m
-x t
l-t
exp(
(6.4)
RECURRENCE RELATIONS.
In this section, we obtain pure and mixed recurrence relations for the polynomials
Fm(x,
n
).
7.1. DIFFERENTIAL RECURRENCE RELATIONS.
In the last section, we have obtained the
Fm(x,%
n
generating function for the polynomials
F
n=0
t
TM
mn+(x’%’ v)
n
e
x
’
t
) given by (6 2):
0FI
x
2+ I + m
im
t
So,
n=0
t
x-Fmmn+v(x’%’) T
e
t
m
0FI
(7.1)
-x
+
2+ %
m
im
Let
G(x,t)
e
t
(--;
0FI
2+l+m- i
m
x t)
m
(7.2)
Differentiating (7.2) partially with respect to x and t and writing G in place of
G(x,t), we get
G
--x
t
m-i
=-m x
t e
0FI’
(--;
2v
+ I +
1
m
m
x t)
m
(7.3)
and
G
t
e
t
0FI(--;
+
2+ %
m- i
m
x t)
m
m
x e
t0F" (--; 2+ % +
1
m- i
m
m
-x t), (7.4)
where prime denotes the differentiation with respect to the argument.
Eliminating
0FI, 0FI
from (7 2), (7 3) and (7.4), we see that G(x,t) satisfies
the following partial differential equation:
x
G
x
m t
8G
Substituting G from (7.1)
--lmn+ (x
[-)X
X
n= 0
)
))+X-%
mn (x
+
(7.5)
m t G.
.
n
)]
-t
m t
n=O
m t
n=0
nx- mran+ (x,
m
x- Fmn+(x’%’
)
t
)
t
n-i
n.l
n
(7.6)
CLASS OF ORTHOGONAL POLYNOMIALS OF A NEW TYPE
177
or
+
mn+v(x,%, x;)
n=0
m
n=O
m
mn+ v(x’
mn+u(x’%’ )
n
n=O
Equating the coefficients of t
D F
x
n
,
t
x;)
m(n-l)+(x
m
n=l
%, ) (n
i)!
from both sides, for n -> I, we obtain the differential
recurrence relation
m
mn+(x’%
xD F
m
m
( + mn) Fmn+(x %, ) + m n F
)
m(n-l)+(x’ I
v)
(7.7)
0
To find out the other differential recurrence relation, let us start from the
generating relation given by (6.4), which can be written as:
2+l+m-i
n=O
m
n
x-
F
m
mn+ (x % V)
2+%+m- i
n
t
m
m
t)
(i
exp(
(7.8)
I- t
Let
2+l+m- i
G(x,t)
m
(i- t)
-x
i
exp
m
t
(7.9)
Writing G in place of G(x,t), and differentiating (7.9) partially with respect to x:
m-i
-m x
t
G
x
(i
(7.10)
G.
t)
Substituting G from (7.8), we obtain
2+ I + m- i
)n
--X
--i
m-I
-m x
m
F
mn+(x’l’ ) +
(i- t)
+
2+ I
t
x-
O F
i
m
n
m
n=0
m
mn+(x
x-
I, )
]
t
n
m
t
I
mn+(x’ )
F
n
or
2+l+m- i
m
)n
2+ I + m
1
n-I
m
m x
m-i
Z
n=l
Equating the coefficients of t
relation
mn+(x %’ v) +
F
x
n
E
2+
m
D F
mn+(x,
, ] tn
)
_Fmm(n_l)+(x,l, )+DFmm(n_l)+(x,l,
)
+
tn
x
m
m- i
n-i
m
m(n-l)+ (x
F
v)
(n
tn
(n- i)!
i)!
from both sides, we obtain the differential recurrence
178
M. DUTTA AND K. P. MANOCHA
2
(n +
+ %
1
m
FTM
n(+ mxm) F
2v+ %
x(n +
mn+ v( x,%,v)
m
1
D F
m
m
mn+ v(x
% v)
m
v) + x n D F
x,%, v)
re(n-l)+ (
m(n-l)+ (x,%,
(7.11)
0.
From the differential recurrence relation (7.7), we
7.2. PURE RECURRENCE RELATION.
obtain
x D
m
Fmn+(x,%
m
( + mn)
)
Fmn+(x
m
m n F
v)
m(n_l)+
x,,)
(7.12)
which implies that
TM
m(n_l)+(x
x D F
% )
b
+ m(n-l)
F
TM
TM
m(n_l)+(x,%,)- m(n-l) Fm(n_2)+(x,%,)
(7.13)
To obtain the pure recurrence relation, we eliminate the derivatives from (7.11),
(7.12) and (7.13), which yields
(n+ 2
+ X
1 ,Fm
mn+(x’ %’ )-(n+
m
n(+
2
+m
%
[(
1
mxm) Fmm(n_l)+(x,%,)
+
ran)
Fmmn+(x,%
+ m(n
n
)
TM
)-mn F
m(n-i )+ (x
i
) Fmm(n_l)+(x,%,)
m
m(n_2)+v(x,%, v)
ran(n- i) F
0.
Hence we obtain
(n
+2+m %-
1
TM
F
mn+V(x’
x Fm
m(n_l)+o(x
)
1
n
+ 2v+m %
% ) + (n- l) F
1
m
m
m(n_2)+(x,,)
0.
(7.14)
In the above equation we see that for a fixed m and for different values of
0,1,2,...,m-1), we shall obtain m different recurrence relations.
Also we are
getting different recurrence relations for different partitions for the set of values
2, then for v
For example if we take m
of the variable parameter n.
polynomials and for
us the recurrence relation for even
0 (7.14) gives
1 it gives us that for odd
polynomials, which is interesting and seems to be new.
8.
SPECIAL CASES.
CASE i.
If we put m
0 and %
i,
1
+
,
then the differential equation
(i.i) reduces to that for Laguerre polynomials as:
xy" + (i +
x)y’ + ny
0.
By the same substitution, relation (4.2) reduces to
FI(x
n
+
,
0)
n!
()
L
(I + a)n n
(x),
(8.1)
CLASS OF ORTHOGONAL POLYNOMIALS OF A NEW TYPE
where L
()
179
(x) is the Laguerre polynomials.
But for %
i,
F
l(x,1
n
L0n
O)
(x)
L (x)
n
i.e. simple Laguerre polynomials.
Also for m
O, and using the
i,
relation
(8.1), the Rodrigues’ formula (5.1
the generating functions given by (6.2) and (6.4) and recurrence relations (7.7),
For I
(7.11) and (7.14) reduce to that for Laguerre polynomials.
I, these results
reduce to that for simple Laguerre polynomials.
CASE 2.
2, %
On putting m
0 differential equation (i.i) becomes
y"
2xy’ + 2ny
O,
which is the differential equation for Hermite polynomials.
By the same substitution, relation (3.1) gives us,
2
(x,0,O)
n
(-1
F
i
n/2
)n/2
n
2Fo (-
n
-+
i
i
2
x
)n/2
2
H (x),
n
n
where H (x) is the Hermite polynomial (see p. 108 of Rainville [8]).
n
we put m
0, v
2, %
F
and for m
2, %
0,
2
where
H2n(X), H2n+l(x)
Once again, if
0, in (4.1) then,
2
(x,O,O)
2n
F2n+l (x
2
n/2
(_i)
I
n
x
IFI
(-n;
F
(-n;
1
n
2
(-i) n’
(2n)!
2
(-i) n’
2(2n + i)!
x
H2n (x)
i,
0 i)
x
1 1
3
x
n
H2n+l(X);
are even and odd Hermite polynomials, see Erdelyi
[i], p. 194.
With the help of these relations, we can calculate generating functions,
recurrence relations, etc. for the polynomials
Next, when we put m
nomials of
2, v
H2n(X)
and
H2n+l(X).
0 in (4.2), it reduces to generalized Hermite poly-
Dutta, Chatterjea and Ghosh [3].
F
2
(x,% O)
2n
n!
% + I
(-----)n
L
(%-1
2
n
(x2)
M. DUTTA AND K. P. MANOCHA
180
(-i)
2,
and for m
=
n
H2n (x)
i, it reduces to:
2
F2n+l
(x,,l)
I+l)
xn!
+ 3
n
HA2n(X)
and
H2n+l%
2
2
(-i)
X + 3
Z
where
(x
Ln
I
)n
H2n+l(X)
(x) are defined in Dutta, Chatterjea and Ghosh [3].
REFERENCES
et al. Higher Transcendental Functions, Vol. II, McGraw-Hill, New
York, Toronto and London, 1953.
i.
ERDELYI, A.
2.
DUTTA, M. and MORE, K.L. A New Class of Generalized Legendre Polynomials,
[athematica 7 30 (1965)
33-41.
3.
DUTTA, M., CHATTERJEA, S.K. and GHOSH, B.K.
4.
DUTTA, M., CHATTERJEA, S.K. and MORE, K.L.
On Generalized Hermite Polynomials,
Bull. Math. Soc. Sci. Math. R.S. de Roumanie 12 6__0, No. 4 (1968) 59-65.
Polynomials, Bull. Inst. of
Math.
On a class of Generalized Hermite
Academia Sinica. _3, No. 2 (Dec. 1975).
5.
KRALL, H.L. and FRINK, O.
6.
DUTTA, M. and BHATTACHARYA, S.P.
7.
DUTTA, M. Orthogonality and Differential Equations, Bull. Cal. Math. Soc. 63
(1971), 143-145.
8.
RAINVILLE, E.D.
A New Class of Orthogonal Polynomials:
Polynomials, Trans. Amer. th. Soc. 65 (1949) 100-115.
The Bessel
The Function Space Properly Associated with the
Class of Differential Equations, Bull. Cal. Math. Soc. 62 (1970) 183-188.
Special Functions, MacMillan Company, New York, 1960.