Browsing by Author "Kesarwani, Aashita"

A new generalization of the modified Bessel function of the second kind Kz(x) is studied. Elegant series and integral representations, a differential-difference equation and asymptotic expansions are obtained for it thereby anticipating a rich theory that it may possess. The motivation behind introducing this generalization is to have a function which gives a new pair of functions reciprocal in the Koshliakov kernel cos⁡(πz)M2z(4x)−sin⁡(πz)J2z(4x) and which subsumes the self-reciprocal pair involving Kz(x). Its application towards finding modular-type transformations of the form F(z,w,α)=F(z,iw,β), where αβ=1, is given. As an example, we obtain a beautiful generalization of a famous formula of Ramanujan and Guinand equivalent to the functional equation of a non-holomorphic Eisenstein series on SL2(Z). This generalization can be considered as a higher level analogue of the general theta transformation formula. We then use it to evaluate an integral involving the Riemann Ξ-function and consisting of a sum of products of two confluent hypergeometric functions.

An exact transformation, which we call a \emph{master identity}, is obtained for the series ??n=1?a(n)e?ny for a?C and Re(y)>0. As corollaries when a is an odd integer, we derive the well-known transformations of the Eisenstein series on $\textup{SL}_{2}\left(\mathbb{Z}\right)$, that of the Dedekind eta function as well as Ramanujan's famous formula for ?(2m+1). Corresponding new transformations when a is a non-zero even integer are also obtained as special cases of the master identity. These include a novel companion to Ramanujan's formula for ?(2m+1). Although not modular, it is surprising that such explicit transformations exist. The Wigert-Bellman identity arising from the a=0 case of the master identity is derived too. The latter identity itself is derived using Guinand's version of the Vorono\"{\dotlessi} summation formula and an integral evaluation of N.~S.~Koshliakov involving a generalization of the modified Bessel function K?(z). Koshliakov's integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new transformation involving the sums-of-squares function rk(n). Some results on functions self-reciprocal in the Watson kernel are also obtained.

An exact transformation, which we call the master identity, is obtained for the first time for the series ∑n=1∞σa(n)e-ny for a∈ C and Re(y) > 0. New modular-type transformations when a is a nonzero even integer are obtained as its special cases. The precise obstruction to modularity is explicitly seen in these transformations. These include a novel companion to Ramanujan’s famous formula for ζ(2 m+ 1). The Wigert–Bellman identity arising from the a= 0 case of the master identity is derived too. When a is an odd integer, the well-known modular transformations of the Eisenstein series on SL2(Z) , that of the Dedekind eta function as well as Ramanujan’s formula for ζ(2 m+ 1) are derived from the master identity. The latter identity itself is derived using Guinand’s version of the Voronoï summation formula and an integral evaluation of N. S. Koshliakov involving a generalization of the modified Bessel function Kν(z). Koshliakov’s integral evaluation is proved for the first time. It is then generalized using a well-known kernel of Watson to obtain an interesting two-variable generalization of the modified Bessel function. This generalization allows us to obtain a new modular-type transformation involving the sums-of-squares function rk(n). Some results on functions self-reciprocal in the Watson kernel are also obtained.