arXiv subjects
Publications and source records attributed to Alexander E Patkowski.
The purpose of this paper is to present a collection of interesting generating functions for partitions which have connections to positive definite binary quadratic forms. In establishing our results we obtain some new Bailey pairs.
We explore some integrals associated with the Riesz function and establish relations to other functions from number theory that have appeared in the literature. We also comment on properties of these functions.
We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re(s)<1.$ Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.
We offer some further applications of some Bailey pairs related to some mock theta functions which were established in a recent study. We discuss and offer some double-sum $q$-series, with new relationships among mock theta functions. We also offer a new relationship between the Bailey pair of Bringmann and Kane with that of Andrews.
We expand on a previous study, offering a generalized wave function associated with the parabolic cylinder function and a connection with a two-particle position-space wave function. We also provide an explicit formula for a wave function associated with recent work by the present author and M. Wolf.
This paper presents the methods to utilizing the $s$-fold extension of Bailey's lemma to obtain $spt$-type functions related to the symmetrized rank function $η_{2k}(n).$ We provide the $k=2$ example, but clearly illustrate how deep connections between higher-order spt functions exist for any integer $k>1,$ and provide several directions for possible research. In particular, we present why the function $spt_M^{*}(n),$ the total number of appearances of the smallest parts of partitions where parts greater than the smallest plus $M$ do not occur, is an $spt$ function that appears to have central importance.
In this paper we offer some new identities associated with mock theta functions and establish new Bailey pairs related to indefinite quadratic forms. We believe our proof is instructive use of changing base of Bailey pairs, and offers new information on some Bailey pairs that have proven important in the study of Mock theta functions.
We discuss some properties of integrals associated with the free particle wave packet, $ψ(x,t),$ which are solutions to the time-dependent Schr$\ddot{o}$dinger equation for a free particle in one dimension. Some noteworthy discussion is made in relation to integrals which have appeared in the literature. We also obtain formulas for half-integer arguments of the Riemann zeta function.
We prove an interesting symmetric $q$-series identity which generalizes a result due to Ramanujan. A proof that is analytic in nature is offered, and a bijective-type proof is also outlined.
Expressions for a family of integrals involving the Hurwitz zeta function are established using standard properties of the Fourier transform.
In this note, we offer some relations and congruences for an interesting $spt$-type function.
We offer a more general Bailey pair than one that was proved in two different papers by two different methods [5, 12].
We examine the $q$-Pell sequences and their applications to weighted partition theorems and values of $L$-functions. We also put them into perspective with sums of tails.