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Michael J. Schlosser

Publications and source records attributed to Michael J. Schlosser.

At least 19 recordsLinked to original sources
arXiv2026

Signed Generalized Stirling Polynomials, Nested Sums, and Hyperbolic Secant Integral Identities

We study signed generalized Stirling polynomials $P_k(m,x)$ arising in closed forms for Malmsten-type hyperbolic secant integrals. Their product structure is used to prove recurrences, gamma--polygamma formulas for $P_{m-s}(m,x)$, a central vanishing identity, a finite approximation to $\cosh πx$, and a limit formula for $π$. We identify these polynomials as signed residues of the equal-period Barnes multiple zeta function, derive their reflection formula, and obtain finite parity-cancellation and Stirling cycle-number identities. We also compare the centered polynomials with a classical Meixner--Pollaczek orthogonal family. We evaluate finite nested sums built from the sequence $χ_n$. Fixing the lower bounds turns these sums into coefficient-counting problems: common lower bounds give binomial coefficients and staircase bounds give Catalan numbers. Combining these counts with known formulas for $χ_j$ yields explicit hyperbolic-secant integral evaluations involving Catalan's constant, zeta values, and polygamma values. A Wolfram Language package accompanies the formulas.

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arXiv2026

A local bijection between alternating sign matrices and descending plane partitions and a Striker--Fulmek-type $q$-statistic

We prove a local bijection between alternating sign matrices (ASMs) and descending plane partitions (DPPs) in the refined fibres of Behrend--Di Francesco--Zinn-Justin. The map is constructed by moving a boundary line through the ASM and DPP pictures. At each step only a bounded neighbourhood of that boundary is changed. In growth-diagram language, the boundary states are ordinary ASM row frontiers and shifted DPP path frontiers, and the elementary moves are reversible Fomin-type square rules. The proof checks that overlapping local moves are confluent, that every square is invertible, and that the completed boundary again satisfies the ordinary ASM and DPP coordinate inequalities. The only terminal choice is resolved by a radius-two fan rule on the visible height contour. The same local framework gives a statistic on refined ASMs whose generating function is the DPP degree enumerator. The statistic has a Striker--Fulmek quadratic core \[ Q_{\rm SF}(A)= \sum_{1\le r<s\le n}\ \sum_{1\le b<a\le n} (n-r+1)A_{r,a}A_{s,b}, \] plus a bounded fan-boundary correction and, outside the normal boundary gauge, an explicit compact-potential endpoint term. In the normal gauge this is \[ W_{\rm ASM}^{\rm SF}(A)=Q_{\rm SF}(A)+B_{\rm fan}(A). \] Thus the paper gives both a local ASM--DPP bijection and a statistic-preserving refined \(q\)-enumeration. The supplementary files reproduce selected local checks and the displayed arithmetic examples.

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arXiv2026

Bilateral $q$-ultraspherical functions

We introduce the bilateral $q$-ultraspherical functions, a bilateral-series extension of the continuous $q$-ultraspherical polynomials. They are defined by specific bilateral basic hypergeometric ${}_2ψ_2$ series, are analytic in the variable $x=\cosθ$, and depend on two parameters $β$ and $γ$ and on a base $q$. We derive a product formula for their bilateral generating function, a three-term recurrence relation, their transformation under the Askey--Wilson divided difference operator, three weight-based Rodrigues-type formulae, and explicit large-order asymptotic expansions. The main results are full orthogonality relations with respect to explicit orthogonality functionals involving analytic mass aggregates. We also obtain shifted orthogonality relations and a bilateral Chen--Liu type mixed orthogonality formula. In the limit $γ\to1$, the construction and identities reduce to the classical results for the continuous $q$-ultraspherical polynomials.

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arXiv2026

Well-poised basic q-Taylor expansions with complementary remainders and a two-basis kernel

We prove a nonterminating well-poised basic $q$-Taylor expansion with complementary remainders for a two-basis infinite-product kernel implicitly proposed by the second author in \cite[Sec.~5]{Schlosser2008}. The well-poised parameter $c$ gives the rational $p=0$ basis, while the elliptic nome $p$ is a separate deformation; the infinite expansions treated here are specific to the basic case. We compute the two Taylor coefficient families and show that each one-family Taylor remainder tends to the complementary basis contribution. The proof uses the well-poised Cooper formula, Jackson's terminating ${}_8ϕ_7$ summation, Rogers' ${}_6ϕ_5$ summation, and theta interpolation, but not Bailey's nonterminating ${}_8ϕ_7$ summation, which is recovered as a consequence. We also record two quadratic one-family examples and discuss a multi-kernel outlook.

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arXiv2026

Transformations and summations for bilateral basic hypergeometric series

We review and derive transformation and summation formulas for bilateral basic hypergeometric series. Our study focuses on consequences of certain bilateral extensions of two important results by Bailey, namely a transformation for very-well-poised $_8W_7$ series in terms of two balanced $_4ϕ_3$ series, and a transformation connecting three $_8W_7$ series. Two rather recently discovered transformations of bilateral basic very-well-poised ${}_8Ψ_8$, one by Zhang and Zhang, the other by Wei and Yu, serve as the starting point of our investigations. From these transformations we work out interesting special cases that were not considered before, including explicit bilateral quadratic and cubic summations. We further explicitly record noteworthy lower-level transformations derived by taking suitable limits and deduce more transformations by exploiting the symmetry of the parameters in the series.

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arXiv2025

Diagonal operators, $q$-Whittaker functions and rook theory

We discuss the problem posed by Bender, Coley, Robbins and Rumsey of enumerating the number of subspaces which have a given profile with respect to a linear operator over the finite field $\mathbb{F}_q$. We solve this problem in the case where the operator is diagonalizable. The solution leads us to a new class of polynomials $b_{μν}(q)$ indexed by pairs of integer partitions. These polynomials have several interesting specializations and can be expressed as positive sums over semistandard tableaux. We present a new correspondence between set partitions and semistandard tableaux. A close analysis of this correspondence reveals the existence of several new set partition statistics which generate the polynomials $b_{μν}(q)$; each such statistic arises from a Mahonian statistic on multiset permutations. The polynomials $b_{μν}(q)$ are also given a description in terms of coefficients in the monomial expansion of $q$-Whittaker symmetric functions which are specializations of Macdonald polynomials. We express the Touchard--Riordan generating polynomial for chord diagrams by number of crossings in terms of $q$-Whittaker functions. We also introduce a class of $q$-Stirling numbers defined in terms of the polynomials $b_{μν}(q)$ and present connections with $q$-rook theory in the spirit of Garsia and Remmel.

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arXiv2025

An Algebra of Elliptic Commuting Variables and an Elliptic Extension of the Multinomial Theorem

We introduce an algebra of elliptic commuting variables involving a base $q$, nome $p$, and $2r$ noncommuting variables. This algebra, which for $r=1$ reduces to an algebra considered earlier by the author, is an elliptic extension of the well-known algebra of $r$ $q$-commuting variables. We present a multinomial theorem valid as an identity in this algebra, hereby extending the author's previously obtained elliptic binomial theorem to higher rank. Two essential ingredients are a consistency relation satisfied by the elliptic weights and the Weierstrass type $\mathsf A$ elliptic partial fraction decomposition. From the elliptic multinomial theorem we obtain, by convolution, an identity equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel-Turaev ${}_{10}V_9$ summation. Interpreted in terms of a weighted counting of lattice paths in the integer lattice $\mathbb Z^r$, this derivation of Rosengren's $\mathsf A_r$ Frenkel-Turaev summation constitutes the first combinatorial proof of that fundamental identity.

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arXiv2025

On an identity of Chaundy and Bullard. III. Basic and elliptic extensions

The identity by Chaundy and Bullard expresses $1$ as a sum of two truncated binomial series in one variable where the truncations depend on two different non-negative integers. We present basic and elliptic extensions of the Chaundy--Bullard identity. The most general result, the elliptic extension, involves, in addition to the nome $p$ and the base $q$, four independent complex variables. Our proof uses a suitable weighted lattice path model. We also show how three of the basic extensions can be viewed as Bézout identities. Inspired by the lattice path model, we give a new elliptic extension of the binomial theorem, taking the form of an identity for elliptic commuting variables. We further present variants of the homogeneous form of the identity for $q$-commuting and for elliptic commuting variables.

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arXiv2025

A family of orthogonal functions on the unit circle and a new multilateral matrix inverse

Using Bailey's very-well-poised $_6ψ_6$ summation, we show that a specific sequence of well-poised bilateral basic hypergeometric $_3ψ_3$ series form a family of orthogonal functions on the unit circle. We further extract a bilateral matrix inverse from Dougall's ${}_2H_2$ summation which we use, in combination with the Pfaff--Saalschütz summation, to derive a summation for a particular bilateral hypergeometric $_3H_3$ series. We finally provide multivariate extensions of the bilateral matrix inverse and the $_3H_3$ summation in the setting of hypergeometric series associated to the root system $A_r$.

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arXiv2025

Residue class biases in unrestricted partitions, partitions into distinct parts, and overpartitions

We prove specific biases in the number of occurrences of parts belonging to two different residue classes $a$ and $b$, modulo a fixed non-negative integer $m$, for the sets of unrestricted partitions, partitions into distinct parts, and overpartitions. These biases follow from inequalities for residue-weighted partition functions for the respective sets of partitions. We also establish asymptotic formulas for the numbers of partitions of size $n$ that belong to these sets of partitions and have a symmetric residue class bias (i.e., for $1\le a<m/2$ and $b=m-a$), as $n$ tends to infinity.

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arXiv2023

Expansions of averaged truncations of basic hypergeometric series

Motivated by recent work of George Andrews and Mircea Merca on the expansion of the quotient of the truncation of Euler's pentagonal number series by the complete series, we provide similar expansion results for averages involving truncations of selected, more general, basic hypergeometric series. In particular, our expansions include new results for averaged truncations of the series appearing in the Jacobi triple product identity, the $q$-Gauß summation, and the very-well-poised ${}_5ϕ_5$ summation. We show how special cases of our expansions can be used to recover various existing results. In addition, we establish new inequalities, such as one for a refinement of the number of partitions into three different colors.

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arXiv2023

An esoteric identity with many parameters and other elliptic extensions of elementary identities

We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can be obtained from telescoping. So we used telescoping in our study to find elliptic extensions of these identities. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case. In addition, we recover some $q$-identities due to Warnaar.

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arXiv2023

Bilateral identities of the Rogers--Ramanujan type

We derive by analytic means a number of bilateral identities of the Rogers--Ramanujan type. Our results include bilateral extensions of the Rogers--Ramanujan and the Göllnitz-Gordon identities, and of related identities by Ramanujan, Jackson, and Slater. We give corresponding results for multiseries including multilateral extensions of the Andrews--Gordon identities, of the Andrews--Bressoud generalization of the Göllnitz--Gordon identities, of Bressoud's even modulus identities, and other identities. Our closed form bilateral and multilateral summations appear to be the very first of their kind.

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arXiv2023

A weighted extension of Fibonacci numbers

We extend Fibonacci numbers with arbitrary weights and generalize a dozen Fibonacci identities. As a special case, we propose an elliptic extension which extends the $q$-Fibonacci polynomials appearing in Schur's work. The proofs of most of the identities are combinatorial, extending the proofs given by Benjamin and Quinn, and in the $q$ case, by Garrett. Some identities are proved by telescoping.

math.CO↗
arXiv2022

Lattice paths and negatively indexed weight-dependent binomial coefficients

In 1992, Loeb considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub showed that these results can be extended to the $q$-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining arithmetic properties of the $q$-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by the second author, to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, $q$- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.

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arXiv2021

Supercongruences involving Domb numbers and binary quadratic forms

In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as congruences involving specialized Bernoulli polynomials, harmonic numbers, binomial coefficients, and hypergeometric summations and transformations.

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arXiv2021

On the infinite Borwein product raised to a positive real power

In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^δ$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $δ$. We use the Hardy--Ramanujan--Rademacher circle method to give an asymptotic formula for the coefficients. For $p=3$ we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent $δ$ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the $p=3$ case.

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