Internat. J. Math. & Math. Sci.
VOL. 12 NO. 3 (1989) 583-588
583
SOME SERIES WHOSE COEFFICIENTS INVOLVE THE VALUES ’(n) FOR n ODD
L.R. BRAGG
Department of Mathematical Sciences
Oakland University
Rochester, Michigan
48309
(Received September I0, 1986)
By using two basic formulas for the digamma function, we derive a variety
that involve as coefficients the values (2n + I), n
1,2,..., of the
A number of these have a combinatorial flavor which we also
Riemann-zeta function.
ABSTRACT.
of
series
express in a trignometric form for special choices of the underlying variable.
We
briefly touch upon their use in the representation of solutions of the wave equation.
KEY
WORDS
AND
PHRASES.
Digamma
function,
Riemann-zeta
function,
series
representations, trigonometric forms, wave solutions.
1980 AMS SUBJECT CLASSIFICATION CODES. 10H05, 35L05.
1.
INTRODUCTION.
For the Riemann-zeta function
(2n)
(2)2n/(2n)!
IB2nl
(z),
in which
it is well known that
B2n
denotes a Bernoulli number [I].
there are no known analogous closed formulas for the numbers
this
brief paper,
we
(2n + I),
n
However,
1,2,... In
call upon two basic formulas for the digamma function
derive series of polynomials and constants that involve these numbers.
z)
to
An
example of such a series is the following:
[.
nffil
(-I)n+l
4
n
(4n-l)
Sums of this type provide insights about these numbers and the relationships among
Aside from number theoretic aspects, series that involve evaluations of various
them.
zeta functions play a role in the foundations of comblnatorlcs
[2].
The formulas we
derive involve polynomials that permit connections with solutions of the wave equation
for certain types of singular data.
While such representations have little practical
value, they illustrate how an intrinsfcally arithmetic function assumes a meaningful
role in a physical problem.
584
L.R. BRAGG
In section 2, we make use of the formulas
I
(a)
(-l)n(n)zn-I
-y +
+ z)
n;2
z
(b)
+ 1)
1/z
z)
(See [3]) to develop a pair of infinite series identities, in a lens shaped
convergence region, that involve the values (n) as well as certain polynomials. The
one of these of interest to us involves only the values (2n + I).
By using
differentiation
and
integration
of
properties
the
polynomials
entering
derive a number of related series that have a combinatoric flavor.
+ iy, one can express these
complex variable has the form z
we
series
in terms
of
for obtaining certain evaluations
trignometric functions which are more convenient
(such as (I.I)).
these,
When the basic
We do this in section 3 and then note the connections with wave
solutions in section 4.
While
the
results
obtained
appear
be
to
can
constructed
corresponding
for
We
developed.
be
easily
results
series
tools
mathematica
used
Further relationships of the type
leave
the
involving
the
novel,
involve little beyond elementary complex variables.
to
it
the
values (2n) by
reader
to
develop
using
the
formula
(2.4b) obtained in the following section.
2.
BASIC SERIES IDENTITIES.
We first define polynomial sets
f
nz
z
2n
{fn(Z)}
and
n(Z)
by means of the relations
_(l_z)2n
(2.1)
2n+l +
z
gn(Z)
for n
0 and take
Using
(l.2a)
(l_z)2n+1
gn(Z)
express I
fn(Z)
to
0 for n
<
O.
+ (l-z)) and l-z) in powers
of
(l-z)
and
(-z)
respectively, we have, by (l.2b),
$(2-z) -(1-z)
$(1 + (-z))
$(1 + (-z))
(2.2)
[ gn(Z)
n=O
$(I+z)
n=O
(2n + 2) +
(z)
gn(Z)
(+z)
g (2n + 2)
[ fn(Z)
n=l
.
(2n + I)
I/(l-z)
Similarly, (1.2) also gives
$( + (z-l))
n=l
(2.3)
fn(Z)
(2n + I)
i/z
SERIES WHOSE COEFFICIENTS INVOLVING THE VALUES OF THE ZELTA FUNCTION
for z e R.
Upon adding (subtracting) the last two members of (2.3) to (from) the
last two members of
.
n=l
(b)
n---O
for z e R.
585
(2.2), we
obtain
-
f (z) (2n + l)
n
gn(Z)
" ("
(2n + 2)
(l-z
-)
(2.4)
+ 3
These serve as the basic starting series.
use of only the first of these.
The subsequent discussion makes
From (2.1), it readily follows that
(a)
D 2p f (z)
(b)
Dz2p+Ifn(Z)
z
[(2n)! / (2n- 2p)! f
n
n-p
(z)
(2.5)
Using these,
follows
it
[(2n)!
I
n;I
(b)
2n +
2p
.
=.
n=
(2n- 2p- I)’].
gn_p_l(Z)
that if we differentiate (2.4a)
respect to z, we get, for z
()
I
2p and 2p +
times w/th
R,
2p) fn(Z)
(2n + 2p + I)
[(i-z) -(2p + I)
z-(2p
+
I)]
(2.6)
+ 2p
(2n 2p+
+ 2
gn(Z)
(2n + 2p + 3)
[(i-z) -(2p + 2) + z -(2p + 2)
q,
if
we multiply (2.4a)
Similarly,
by zP(l-z)
p,q positive integers, and
integrate with respect to z from 0 to I, we get using the definition of the beta
function,
{B(2n + p + I, q + I) -B(p + I, 2n + q + I)} (2n + I)
n=l
=5
(2.7)
[B(p + I, q)- B(p, q + 1)]
One can obtain formulas analogous to (2.7) by using (2.6).
For z e R, the formulas (2.6) are convenient for obtaining a number of specific
For example, if we take z
I/2 in (2.6b), we get
series.
n=0
-"
2n + 2p +
2p +
2)
(2n + 2p + 3)
22p
+ 2
(2.8)
586
L.R. BRAGG
Similarly,
VL
)
in
(2.7) and simplify the corresponding beta
I,
(2n + 1)
(2n + 2p + 2)
n
-(2n +
n=l
3.
p +
q
select
we
if
functions, we have for p
__ip( -1)!
p + I) (2n + p + 2)
(2p + I)’
(2.9)
TRIGNOMETRIC FORMS.
I/2 + iy with
Suppose z
tan
l(2y),
ly <
3/2
Then z
and (l-z)
Note that
e
2n
2 i p
sin (2n8)
2n+
cos (2n + 1)8
gn(Z) 2
ik O
-ik O.
(l-z) -k +/- z -k
+/- e
(a)
e
181
<
iO
/3
2
+ y2
with p
Then we have
f (z)
n
(b)
(3.1)
p-k[e
(c)
.
Using these in (2.6), we deduce that
Ca)
n=l
(2n 2p+ 2p) o2n
(2n8) (2n + 2p + I)
sin
-(2p + I) sin (2p + 1)8
-
n=O
If we select O
(-I) n+l
[
2
n--1
For p
2n
(3.2)
+ 2p
[ (2n 2p+
(b)
+ 2
p
2n +
-(2p+2) cos (2p+2)0
p
/4, then
4n + 2p2p
I/ 2
0
2)
[ (-l)n
(3) +
Then from (3.2a), we get
(4n + 2p
O, this reduces to (I.I).
gives
cos((2n + 1)O) K(2n + 2p + 3)
I)
2
p-2
/
{sin (2p + I) /4}.
(3.3)
Similarly, this choice for O in (3.2b) with p
[4n (4n + I) + (2n + I) (4n + 3)]
0
(3.4)
n=l
Other choices for 8 (such as /6) and p in the formulas (3.2) will lead to additional
identities.
4.
CONNECTIONS WITH WAVE FUNCTIONS.
l,n(x,t)
2w(x,t)
The wave polynomials w
equation
2w(x,t)
2
2
and
-
W2,n(X,t),
n
0,1,2,..., are solutions of the
(4.1)
that correspond, respectively, to the initial conditions
Wl, n (x’ 0)
0 and w
2 ,n
(x 0)
0
W2,n(X
0)
x
n
Wl,n(X,O)
([4], [5]).
x
n
They are given
SERIES WHOSE COEFFICIENTS INVOLVING THE VALUES OF THE ZELTA FUNCTIONS
587
explicitly by
(a)
Wl, n (x,t) =-2-
(b)
(x t)
[(x + t)
n+
(x-t) n
(4.2)
w2, n
[(x +
2(n+l)
t
)n
+
I_( x-t) n
+
Suppose
If we evaluate (2.4a) first at z
x + t and then at x
t and
add, it follows from
the definition of f (z) and (4.2a) that
n
n=l
The series
in
[Wl,2n(X,t) Wl,2n(l-x,t)](2n
the
+ 1)
l-x
(l_x)2_t
x
2
x2_t 2]
left member of this converges in the interior S
of
(4.3)
the square
Note that the right
(0,0), (I,0), (1/2, /2), and (I/2,
I/x) at t
0 hich has singularities at x
member of (4.3) reduces to
0 and
1.
x
he characteristics of the equation (4.1) through he points (0,0) and (1,0)
The formula (2.6a) can be used to construct other such
determine this same region S
sries involving the (2n + I). Finally, one can use (2.4b) to construct examples of
of
series
the
wave
that
the
values (2n) as
involve
polynomials
2/2).
having vertices at
W2,n(X’t)
coefficients.
REFERENCES
I.
MAGNUS, W., OBERHETTINGER, F. and SONI, R., Formulas and theorems for special
functions of mathematical physics, Springer-Verlag, New York (1966).
2.
AIGNER, M., Combinatorial theory, Springer-Verlag, New York (1979).
ABRAMOWITZ, M. and STEGUN, I.A., Ed. Handbook of mathematical functions,
National Bur. of Standards, Washington, D.C. (1968).
BRAGG, L.R. and DETTMAN, J.W., Expansions of solutions of certain hyperbolic
and elliptic problems in terms of Jacobi polynomials, Duke Math. J. 36(1969),
3.
4.
129-144.
5.
WIDDER, D.V., Expansions in
series of homogeneous polynomial solutions of the twodimensional wave equation, Duke Math. J. 26(1959), 591-598.