TY - JOUR

T1 - Two general series identities involving modified Bessel functions and a class of arithmetical functions

AU - Berndt, Bruce C.

AU - Dixit, Atul

AU - Gupta, Rajat

AU - Zaharescu, Alexandru

N1 - Publisher Copyright:
© 2022 Cambridge University Press. All rights reserved.

PY - 2023/12

Y1 - 2023/12

N2 - We consider two sequences a(n) and, b(n), 1 ≤ n < ∞ generated by Dirichlet series {equation presented} satisfying a familiar functional equation involving the gamma function Γ(s). Two general identities are established. The first involves the modified Bessel function Kμ(z), and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are Kμ(z), the Bessel functions of imaginary argument Iμ(z), and ordinary hypergeometric functions 2F1(a,b;c;z). Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan's arithmetical function τ(n), the number of representations of n as a sum of k squares rk(n), and primitive Dirichlet characters Χ(n).

AB - We consider two sequences a(n) and, b(n), 1 ≤ n < ∞ generated by Dirichlet series {equation presented} satisfying a familiar functional equation involving the gamma function Γ(s). Two general identities are established. The first involves the modified Bessel function Kμ(z), and can be thought of as a 'modular' or 'theta' relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are Kμ(z), the Bessel functions of imaginary argument Iμ(z), and ordinary hypergeometric functions 2F1(a,b;c;z). Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan's arithmetical function τ(n), the number of representations of n as a sum of k squares rk(n), and primitive Dirichlet characters Χ(n).

KW - Bessel functions

KW - classical arithmetic functions

KW - functional equations

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U2 - 10.4153/S0008414X22000530

DO - 10.4153/S0008414X22000530

M3 - Article

AN - SCOPUS:85140229715

SN - 0008-414X

VL - 75

SP - 1800

EP - 1830

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

IS - 6

ER -