arXiv subjects

Maxie D. Schmidt

Publications and source records attributed to Maxie D. Schmidt.

22 records · Page 2Linked to original sources

arXiv2017

New Congruences and Finite Difference Equations for Generalized Factorial Functions

We use the rationality of the generalized $h^{th}$ convergent functions, $Conv_h(α, R; z)$, to the infinite J-fraction expansions enumerating the generalized factorial product sequences, $p_n(α, R) = R(R+α)\cdots(R+(n-1)α)$, defined in the references to construct new congruences and $h$-order finite difference equations for generalized factorial functions modulo $h α^t$ for any primes or odd integers $h \geq 2$ and integers $0 \leq t \leq h$. Special cases of the results we consider within the article include applications to new congruences and exact formulas for the $α$-factorial functions, $n!_{(α)}$. Applications of the new results we consider within the article include new finite sums for the $α$-factorial functions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers $h \geq 2$.

math.CO↗

arXiv2016

Zeta Series Generating Function Transformations Related to Generalized Stirling Numbers and Partial Sums of the Hurwitz Zeta Function

We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within the article satisfy expansions by generalized harmonic number sequences, or the partial sums of the Hurwitz zeta function, which are analogous to known properties for the Stirling numbers of the first kind and for the known transformation coefficients employed to enumerate variants of the polylogarithm function series. Applications of the new results we prove in the article include new series expansions of the Dirichlet beta function, the Legendre chi function, BBP-type series identities for special constants, alternating and exotic Euler sum variants, alternating zeta functions with powers of quadratic denominators, and particular series defining special cases of the Riemann zeta function constants at the positive integers $s \geq 3$.

math.CO↗

arXiv2016

Generating Function Transformations Related to Polylogarithm Functions and the $k$-Order Harmonic Numbers

We define a new class of generating function transformations related to polylogarithm functions, Dirichlet series, and Euler sums. These transformations are given by an infinite sum over the $j^{th}$ derivatives of a sequence generating function and sets of generalized coefficients satisfying a non-triangular recurrence relation in two variables. The generalized transformation coefficients share a number of analogous properties with the Stirling numbers of the second kind and the known harmonic number expansions of the unsigned Stirling numbers of the first kind. We prove a number of properties of the generalized coefficients which lead to new recurrence relations and summation identities for the $k$-order harmonic number sequences. Other applications of the generating function transformations we define in the article include new series expansions for the polylogarithm function, the alternating zeta function, and the Fourier series for the periodic Bernoulli polynomials. We conclude the article with a discussion of several specific new "almost" linear recurrence relations between the integer-order harmonic numbers and the generalized transformation coefficients, which provide new applications to studying the limiting behavior of the zeta function constants, $ζ(k)$, at integers $k \geq 2$.

math.CO↗

arXiv2016

A Computer Algebra Package for Polynomial Sequence Recognition

The software package developed in the MS thesis research implements functions for the intelligent guessing of polynomial sequence formulas based on user-defined expected sequence factors of the input coefficients. We present a specialized hybrid approach to finding exact representations for polynomial sequences that is motivated by the need for an automated procedures to discover the precise forms of these sums based on user guidance, or intuition, as to special sequence factors present in the formulas. In particular, the package combines the user input on the expected special sequence factors in the polynomial coefficient formulas with calls to the existing functions as subroutines that then process formulas for the remaining sequence terms already recognized by these packages. The factorization-based approach to polynomial sequence recognition is unique to this package and allows the search functions to find expressions for polynomial sums involving Stirling numbers and other special triangular sequences that are not readily handled by other software packages. In contrast to many other sequence recognition and summation software, the package not provide an explicit proof, or certificate, for the correctness of these sequence formulas -- only computationally guided educated guesses at a complete identity generating the sequence over all $n$. The thesis contains a number of concrete, working examples of the package that are intended to both demonstrate its usage and to document its current sequence recognition capabilities.

math.CO↗