Integral Representation of Sums of Series Associated with Special Functions
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Integral Representation of Sums of Series Associated with Special Functions K. A. Mirzoev1* and T. A. Safonova2** 1 2
Lomonosov Moscow State University, Moscow, 119991 Russia
Northern Arctic Federal University, Arkhangelsk, 163002 Russia
Received April 12, 2020; in final form, April 12, 2020; accepted May 14, 2020
DOI: 10.1134/S0001434620090369 Keywords: Green function of ordinary differential operators, integral representation of sums of series, Riemann zeta function, Dirichlet beta function, Catalan and Apery ´ constants.
1. Following [1, Chap. 23], by β(n) and η(n) we denote the sums β(n) =
+∞ (−1)k−1 , (2k − 1)n
η(n) =
k=1
+∞ (−1)k−1 k=1
kn
,
n = 1, 2, . . . .
The function β(n) is called the Dirichlet beta function (see, e.g., [2, Chap. 1, Sec. 1.7]); it is well known that β(2n + 1) =
(−1)n (π/2)2n+1 E2n , 2(2n)!
n = 0, 1, . . . .
It is also known that η(1) = ln 2, η(n) = (1 − 21−n )ζ(n), n = 2, 3, . . . , where ζ(n) is the Riemann zeta function, and η(2n) =
(−1)n−1 (1 − 21−2n )(2π)2n B2n , 2(2n)!
n = 1, 2, . . . .
As usual, the En and Bn denote the Euler and Bernoulli numbers (see [1, Chap. 23, formulas 23.2.22 and 23.2.16]; see also [2]–[5]). Recall that the numbers β(2)(= G) and ζ(3) are customarily called the Catalan constant and the Apery ´ constant, respectively (see, e.g., [2, Chap. 1, Secs. 1.7 and 1.6]). In Sec. 2 of this paper, using methods proposed by the authors in their earlier papers [6] and [7], we obtain integral representations of the sums of the series +∞ k=1
(−1)k−1 , (2k − 1)2 ± a2
+∞ (−1)k−1 , k(k2 ± a2 ) k=1
where −1 < a < 1 (see Theorems 1 and 2); obviously, the first two of them are generating functions for the numbers β(2n) and the next two, for η(2n − 1) and, therefore, for ζ(2n + 1), n = 1, 2, . . . . Section 3 is devoted to the explicit calculation of the integrals obtained in Theorem 1 of Sec. 2 in the case where the parameter a is a rational number (see Lemma 1 and Theorem 3); in Sec. 4, these integrals are calculated in terms of hypergeometric series for any “admissible” a ∈ C. Special attention is given to the application of the obtained results to various representations of the ´ constants and of ln 2. Catalan and Apery * **
E-mail: [email protected] E-mail: [email protected]
617
618
MIRZOEV, SAFONOVA
The subject of this paper has long been the center of attention of many mathematicians and has an extensive literature. In particular, in the papers [8] and [9], transformation formulas for sums of the form +∞ k=1
1 , 2 k − a2
+∞ k=1
1 , 2 k(k − a2 )
+∞ k=1
k4
k − a4
were given and used to obtain representations of the numbers ζ(4n + 3) and ζ(2n + 2) in the form of fairly rapidly converging series. The paper [10] and a substantial part of the recent excellent paper [11] are devoted to the representation of the numbers ζ(2n + 1) in the form ζ(2n + 1) = π 2n
+∞
Rn,k ζ(2k),
k=0
where the coefficients Rn,k are rational functions of k and n. 2. Let −1 < a < 1, and let S be the self-adjoint operator on the Hilbert
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