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 - 2022

Y1 - 2022

N2 - We consider two sequences a(n) and b(n), 1 ≤ n < ∞, generated by Dirichlet series Σ∞ a(n)/λsn and Σ∞ b(n)μns, n=1 n=1 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 x(n).

AB - We consider two sequences a(n) and b(n), 1 ≤ n < ∞, generated by Dirichlet series Σ∞ a(n)/λsn and Σ∞ b(n)μns, n=1 n=1 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 x(n).

KW - Bessel functions

KW - classical arithmetic functions

KW - functional equations

UR - http://www.scopus.com/inward/record.url?scp=85140229715&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85140229715&partnerID=8YFLogxK

U2 - 10.4153/S0008414X22000530

DO - 10.4153/S0008414X22000530

M3 - Article

AN - SCOPUS:85140229715

JO - Canadian Journal of Mathematics

JF - Canadian Journal of Mathematics

SN - 0008-414X

ER -