Multiple Convolution Formulae of Bernoulli and Euler Numbers
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DOI: 10.1007/s13226-020-0444-2
MULTIPLE CONVOLUTION FORMULAE OF BERNOULLI AND EULER NUMBERS Wenchang Chu School of Mathematics and Statistics, Zhoukou Normal University, (Henan), P. R. China and Department of Mathematics and Physics, University of Salento, P. O. Box 193 73100 Lecce, Italy e-mail: [email protected] (Received 17 January 2017; after final revision 10 May 2019; accepted 15 May 2019) By examining higher derivatives of hyperbolic functions, we derive monomial and binomial representation formulae, that are utilized to establish several multiple convolution identities for the Bernoulli numbers Bn , Euler numbers En and two variants of Bn Key words : Bernoulli numbers; Euler numbers; Stirling numbers; hyperbolic functions; multiple convolution. 2000 AMS Subject Classification: 11B68, 05A19.
1. O UTLINE AND I NTRODUCTION The Bernoulli and Euler numbers are generated respectively by the following Maclaurin series (cf. 12, §9.6)
∞
X xn x = Bn ex − 1 n! n=0
∞
and
X xn 2ex = En . 1 + e2x n! n=0
They are well-known in mathematical literature and have wide applications in classical mathematics (see Stromberg [20, Chapters 7] for example). Various sums on Bernoulli and Euler numbers have been considered by Euler, Ramanujan and others. In particular, their convolution formulae have attracted much interest. Two simplest examples due to Euler and Ramanujan (see Berndt [2, Entries 18
970
WENCHANG CHU
& 22]) may be reproduced as follows: m ³ X 2m ´ B2k B2m−2k = (1 − 2m)B2m for m ≥ 2, k=0 2k ¶ m µ X 2m B2m+2 . E2k E2m−2k = 4m+1 (4m+1 − 1) 2m + 2 2k k=0
These identities have been proven in different manners and extended along different directions (cf. [1, 4, 10, 11, 16]). In 1986, Sitaramachandrarao and Davis [19] evaluated the triplicate and quadruplicate convolutions for the Bernoulli numbers. More general problem of computing multiple convolutions for both Bernoulli and Euler numbers was resolved by Dicher [9] in 1996, who succeeded in deriving analytical formulae. Recall that Bernoulli numbers and Euler numbers can be used to express the following Maclaurin series of hyperbolic functions (cf. Stromberg [20, Chapters 5 & 7] and [12, §1.41 & §1.42]) x tanh x = x coth x sechx xcschx
= = =
∞ X (2x)2n n=1 ∞ X n=0 ∞ X n=0 ∞ X n=0
0 B2n ,
(1a)
(2x)2n B2n , (2n)!
(1b)
x2n E2n , (2n)!
(1c)
x2n 00 B ; (2n)! 2n
(1d)
(2n)!
where for brevity, Bn0 and Bn00 are defined respectively by Bn0 := (2n − 1)Bn
and Bn00 := (2 − 2n )Bn .
(2)
We shall investigate, in the present paper, further multiple convolutions of Bernoulli numbers Bn and Euler numbers En as well as two variants Bn0 and Bn00 by examining higher derivatives of the aforementioned hyperbolic functions. As preliminaries, we shall review, in the next section, the explicit expressions for the higher derivative polynomials Pn (y) and Qn (y), and establish the monomial and binomial representation formulae. They will be employed in the third section to derive ten general multiple convolution identities for the four sequences Bn , En , Bn0 and Bn0
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