Error functions, Mordell integrals and an integral analogue of a partial theta function

Dixit, Atul; Roy, Arindam; Zaharescu, Alexandru

doi:10.4064/aa8207-5-2016

Cited by 9 publications

(8 citation statements)

References 15 publications

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“…New modular relations can be obtained in the setting of Koshliakov zeta functions by studying generalizations of the integrals involving the Riemann Ξ-function. A plethora of such integrals in the setting of the Riemann zeta function have been studied in [15], [18], [20] and [21].…”

Section: Discussionmentioning

confidence: 99%

Koshliakov zeta functions I: Modular Relations

Dixit¹,

Gupta²

2021

Preprint

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We examine an unstudied manuscript of N. S. Koshliakov over 150 pages long and containing the theory of two interesting generalizations ζp(s) and ηp(s) of the Riemann zeta function ζ(s), which we call Koshliakov zeta functions. His theory has its genesis in a problem in the analytical theory of heat distribution which was analyzed by him. In this paper, we further build upon his theory and obtain two new modular relations in the setting of Koshliakov zeta functions, each of which gives an infinite family of identities, one for each p ∈ R + . The first one is a generalization of Ramanujan's famous formula for ζ(2m + 1) and the second is an elegant extension of a modular relation on page 220 of Ramanujan's Lost Notebook. Several interesting corollaries and applications of these modular relations are obtained including a new representation for ζ(4m + 3).

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Section: Discussionmentioning

confidence: 99%

Koshliakov zeta functions I: Modular Relations

Dixit¹,

Gupta²

2021

Preprint

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“…Since the variable of integration in ∞ 0 xe −πα 2 x 2 e 2πx − 1 dx runs only from 0 to ∞ similar to the summation indices in the partial theta functions in (1.1.4) running only from 1 to ∞, the integral is called an 'integral analogue of partial theta function' in [20].…”

Section: Introductionmentioning

confidence: 99%

“…where n ∈ R, and represented it in terms of equivalent integrals [48,Equations (19), (20), (21)] 1 . He does not specify any connection whatsoever between the above integral and the corresponding one in (1.1.1).…”

Section: Introductionmentioning

confidence: 99%

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Superimposing theta structure on a generalized modular relation

Dixit¹,

Kumar²

2020

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A generalized modular relation of the form F (z, w, α) = F (z, iw, β), where αβ = 1 and i = √ −1, is obtained in the course of evaluating an integral involving the Riemann Ξ-function. It is a two-variable generalization of a transformation found on page 220 of Ramanujan's Lost Notebook. This modular relation involves a surprising generalization of the Hurwitz zeta function ζ(s, a), which we denote by ζw(s, a). While ζw(s, 1) is essentially a product of confluent hypergeometric function and the Riemann zeta function, ζw(s, a) for 0 < a < 1 is an interesting new special function. We show that ζw(s, a) satisfies a beautiful theory generalizing that of ζ(s, a) albeit the properties of ζw(s, a) are much harder to derive than those of ζ(s, a). In particular, it is shown that for 0 < a < 1 and w ∈ C, ζw(s, a) can be analytically continued to Re(s) > −1 except for a simple pole at s = 1. This is done by obtaining a generalization of Hermite's formula in the context of ζw(s, a). The theory of functions reciprocal in the kernel sin(πz)J2z(2and Jz(x), Yz(x) and Kz(x) are the Bessel functions, is worked out. So is the theory of a new generalization of Kz(x), namely, 1Kz,w(x). Both these theories as well as that of ζw(s, a) are essential to obtain the generalized modular relation.

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“…was obtained. In a recent paper, Dixit, Roy and Zaharescu [2] established an analogous formula for the integral…”

Section: Introductionmentioning

confidence: 99%

A note on an integral of Dixit, Roy and Zaharescu

Paris¹

2018

Preprint

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In a recent paper, Dixit et al. [Acta Arith. 177 (2017) 1-37] posed two open questions whether the integralfor α > 0 could be evaluated in closed form when k is a positive even and odd integer. We establish that Ĵk (α) can be expressed in terms of a Gauss hypergeometric function and a ratio of two gamma functions, together with a remainder expressed as an integral. An upper bound on the remainder term is obtained, which is shown to be exponentially small as k becomes large when a = O(1).

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Error functions, Mordell integrals and an integral analogue of a partial theta function

Cited by 9 publications

References 15 publications

Koshliakov zeta functions I: Modular Relations

Koshliakov zeta functions I: Modular Relations

Superimposing theta structure on a generalized modular relation

A note on an integral of Dixit, Roy and Zaharescu

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