JOURNAL
OF MATIKMATICAL
ANALYSIS
AND
APPLICATIONS
130,
509-5 t 3 (1988)
An Explicit
Formula
for
Generalized
Bernoulli
Polynomials
the
H. M. SRIVASTAVA*
Department
Vicforia,
of Mathematics,
Brifish Columbia
University
of Victoria,
V8W 2Y2, Canada
AND
PAVEL G. TODOROV
Faculty
of Mathematics,
Paissii
Hilendarski
Submitted
Received
University,
by G.-C.
October
BG-4000
Plovdiv,
Bulgaria
Rota
30, 1986
The object of the present note is to prove a new explicit
formula
for the
generalized
Bernoulli
polynomials.
The main result
(3) below
provides
an
interesting
extension
of a representation
for the generalized
Bernoulli
numbers
given recently by P. G. Todorov
[CR.
Arad. Sri. Paris Ser. I Moth. 301 (1985).
665-6661.
c 1988 Academic Press. Inc
1. INTRODUCTION
In the usual notations, let B:)(x) denote the generalized Bernoulli
polynomial of degree n in x, defined by
z
(
1
eZ- 1>
&TX==
2
q=‘(x);
n.
fZ=O
(lzl<27r; l”=
1)
(1)
for an arbitrary (real or complex) parameter ~1.Clearly,
B(‘)(x)
= ( - 1)” B(“)(
n
n tl - x),
so that
B’“‘(cl)
= ( - 1)” B’“‘(O)
Z ( - 1)” B(1)
”
n
n
* The work of this author
Research Council of Canada
was supported,
in part,
under Grant A7353.
by the Natural
Sciences
and Engineering
509
0022-247X/88
$3.00
Copyright
:i 1988 by Academic Press. Inc.
All rights of rcproductmn
in any form reserved.
510
SRIVASTAVAANDTODOROV
in terms of the generalized Bernoulli numbers Bf) (cf. [3, p. 227,
Exercise 181). From the generating relation (1) it is fairly straightforward
to deduce the addition theorem:
B’*+fl)(x+y)=
”
f (2) Bf’(x) BIP_),(y),
k=O
which, for x = p = 0, corresponds to the elegant representation:
for the generalized Bernoulli polynomials as a finite sum of the generalized
Bernoulli numbers.
Recently, Srivastava, Lavoie, and Tremblay [S, p. 442, Eqs. (4.4) and
(4.5)] gave two new classes of addition theorems for the generalized
Bernoulli polynomials. In the present note we first prove the following
explicit formula for these generalized Bernoulli polynomials:
x F[k - n, k-q
2k + l;j/(x
+j)],
(3)
where &‘[a, b; c; z] denotes the Gaussian hypergeometric
by (cf., e.g., 12, Chap. 11)
F[a,b;c;z]=l+TiS+
ab z
a(a+ l)b(b+
c(c+l)
function defined
l)z2
zT+
. ....
(4)
We shall also apply the representation (3) in order to derive certain
interesting special cases considered earlier by Gould [4] and Todorov [6].
2. PROOF OF THE EXPLICIT FORMULA (3)
By Taylor’s
yields
expansion
and Leibniz’s
rule, the generating
Since
(l+w)-“=
f (a+;-l)(-w)~
I=0
(Iwl<l),
relation
(1)
GENERALIZED
BERNOULLI
511
POLYNOMIALS
setting 1 + w = (e’ - 1)/z, and applying the binomial
(5) that
theorem, we find from
B~~(x,=,~o(~)xn-s~o(~+~-‘)
x i (-l)k
k=O
(6)
Now make use of the well-known formula
(e’-
l)k=
f
(cf., e.g., [4, p. 481)
2 A’O’,
(7)
r=k
where, for convenience,
A being the difference operator defined by (cf. [3, p. 13 et
4-(x)
=f(x
seq.])
+ 1) -f(x),
so that, in general,
Akf(x)
= f
(-I)-’
j=O
Formula
; f(x+j).
(7) readily yields
0; {(F>*}
and upon substituting
series, we have
B!;‘(x)=
0
k=O
i (-1)”
s
XC
I=0
I,I,=&Ako‘+k,
(9)
this value in (6), if we rearrange the resulting triple
(~+~-‘):F,*(s:k)(i:+:k)i!x.-‘~*d’o”’*
cr+k+f-1
I
>
(10)
The innermost sum in (10) can be evaluated by appealing to the elementary combinatorial
identity:
512
SRIVASTAVA AND TODOROV
and if we further substitute for dkOSCZk from the definition
we obtain
(8) with a=O,
By(x)=,_,(;)(~+~-l)~xn-*~~(-l)/(:)i2k
f
x F[k-n,
cr+k+
1,2k+
1; -j/x]
(11)
in terms of the Gaussian hypergeometric function defined by (4).
Finally, we apply the known transformation [ 1, p. 3, Eq. (6)]
F[a, b; c; z] = (1 - z) -’ F[a, c - b; c; z/z - 1)],
and (11) leads us immediately
to the explicit formula (3).
3. APPLICATIONS
By Vandermonde’s
F[-N,b;c;
I]=
theorem [2, p. 31, we have
y-‘>(c+;-‘)
’
(N=O,
1,2, ...).
which (for N = n - k, b = k - CI, and c = 2k + 1) readily yields
F[k-n,k-cr;2k+l;
I]=
i
cxfn
n-k
(n-k)! (2k)!
> (n+k)!
(O<kdn).
(12)
In view of (12), the special case of our formula (3) when x = 0 gives us the
following representation for the generalized Bernoulli numbers:
or, equivalently,
in terms of the difference operation exhibited by (8).
Alternatively, since [3, p. 204, Theorem A]
Sfn, k) = h AkO”,
GENERALIZED
BERNOULLI
where S(n, k) denotes the Stirling
[3, p. 207, Theorem B]
513
POLYNOMIALS
number of the second kind, defined by
k! qn, k),
this last representation
By=
(13) or (14) can be written also as
i (-l)k
k=O
(;‘;)(“+;-
‘)(“;“)-’
S(n+k,k).
(15)
Formula (15), given recently by Todorov [6, p. 665, Eq. (3)], provides
an interesting generalization of the following known result for the Bernoulli
numbers B)I s B(l):
”
B,, = 2 ( - 1)k
k=O
(;::)(n:k)-‘y;
which was considered, for example, by Gould [4, p. 49, Eq. (17)].
We should like to conclude by remarking that, in view of Todorov’s
result (15) and the representation (2), it is not diflicult to construct an
alternative proof of our explicit formula (3).
REFERENCES
I. P. APPELL ET J. KAMP~ DE FBRIET, “Fonctions Hypergiom&triques et Hyperspheriques:
PolynBmes d’Hermite,” Gauthier-Villars, Paris, 1926.
2. W. N. BAILEY, “Generalized Hypergeometric Series,” Cambridge Univ. Press, Cambridge/
London/New York, 1935.
3. L. COMTET, “Advanced Combinatorics: The Art of Finite and Inlinite Expansions”
(Translated from the French by J. W. Nienhuys), Reidel, Dordrecht/Boston, 1974.
4. H. W. GOULD, Explicit formulas for Bernoulli numbers, Amer. Math. Monthly 79 (1972),
4451.
5. H. M. SRIVASTAVA, J.-L. LAVOIE,
Math. Bull. 26 (1983), 438445.
AND
R. TREMBLAY, A class of addition theorems, Canad.
6. P. G. TODOROV. Une formule simple explicite des nombres de Bernoulli gSraI&s,
C.R. Acad.
Sci. Paris Se?. I Math.
301 (1985),
665-666.