Mathematics
https://repository.iitgn.ac.in/handle/123456789/547
2022-05-23T21:04:25ZTwo general series identities involving modified bessel functions and a class of arithmetical functions
https://repository.iitgn.ac.in/handle/123456789/7690
Two general series identities involving modified bessel functions and a class of arithmetical functions
Berndt, Bruce C.; Dixit, Atul; Gupta, Rajat; Zaharescu, Alexandru
We consider two sequences a(n) and b(n), 1≤n<∞, generated by Dirichlet series ∑n=1∞a(n)λsnand∑n=1∞b(n)μsn,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).
2022-04-01T00:00:00ZAsymptotics and sign patterns for coefficients in expansions of Habiro elements
https://repository.iitgn.ac.in/handle/123456789/7674
Asymptotics and sign patterns for coefficients in expansions of Habiro elements
Goswami, Ankush; Jha, Abhash Kumar; Kim, Byungchan; Osburn, Robert
We prove asymptotics and study sign patterns for coefficients in expansions of elements in the Habiro ring which satisfy a strange identity. As an application, we prove asymptotics and discuss positivity for the generalized Fishburn numbers which arise from the Kontsevich-Zagier series associated to the colored Jones polynomial for a family of torus knots. This extends Zagier's result on asymptotics for the Fishburn numbers.
2022-04-01T00:00:00ZExtended higher Herglotz function \textup{II}
https://repository.iitgn.ac.in/handle/123456789/7675
Extended higher Herglotz function \textup{II}
Gupta, Rajat; Kumar, Rahul
Very recently, Radchenko and Zagier revived the theory of Herglotz functions. The main goal of the article is to show that one of the formulas on page 220 of Ramanujan's Lost Notebook actually lives in the realms of this theory. As a consequence of our general theorem, we derive an interesting identity analogous to Ramanujan's formula for ?(2m+1). We also introduce a character analogue of the Herglotz function and initiate its theory by obtaining an elegant functional equation governed by it.
2022-04-01T00:00:00ZRaj Chandra Bose: universal mathematician
https://repository.iitgn.ac.in/handle/123456789/7680
Raj Chandra Bose: universal mathematician
Frisinger, H. Howard; Sengupta, Indranath
2022-02-01T00:00:00Z