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Euler's classic partition identity states that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts. We develop a new generalization of this identity, which yields a previous generalization of Franklin as a special case, and prove an accompanying Beck-type companion identity.

Recently, the authors with Lea Beneish established a recipe for constructing Ramanujan-Sato series for $1/\pi$, and used this to construct 11 explicit examples of Ramanujan-Sato series arising from modular forms for arithmetic triangle groups of non-compact type. Here, we use work of Chisholm, Deines, Long, Nebe and the third author to prove a general $p$-adic supercongruence theorem through an explicit connection to CM hypergeometric elliptic curves that provides $p$-adic analogues of these Ramanujan-Sato series. We further use this theorem to construct explicit examples related to each of our explicit Ramanujan-Sato series examples.

The Alder-Andrews Theorem, a partition inequality generalizing Euler's partition identity, the first Rogers-Ramanujan identity, and a theorem of Schur to $d$-distinct partitions of $n$, was proved successively by Andrews in 1971, Yee in 2008, and Alfes, Jameson, and Lemke Oliver in 2010. While Andrews and Yee utilized $q$-series and combinatorial methods, Alfes et al. proved the finite number of remaining cases using asymptotics originating with Meinardus together with high-performance computing. In 2020, Kang and Park conjectured a "level $2$" Alder-Andrews type partition inequality which relates to the second Rogers-Ramanujan identity. Duncan, Khunger, the second author, and Tamura proved Kang and Park's conjecture for all but finitely many cases using a combinatorial shift identity. Here, we generalize the methods of Alfes et al. to resolve nearly all of the remaining cases of Kang and Park's conjecture.

In a 2016 paper, Straub proved an analogue to Euler's partition identity for partitions with fixed perimeter. Later, Fu and Tang provided a refinement and generalization of Straub's analogue to $d$-distinct partitions as well as a result related to the first Rogers-Ramanujan identity. Motivated by Alder-type partition identities and their generalizations, we build on work of Fu and Tang to establish generalized Alder-type partition inequalities in a fixed perimeter setting, and notably, a reverse Alder-type inequality., Comment: 13 pages

In 2020, Kang and Park conjectured a "level $2$" Alder-type partition inequality which encompasses the second Rogers-Ramanujan Identity. Duncan, Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for all but finitely many cases utilizing a "shift" inequality and conjectured a further, weaker generalization that would extend both Alder's (now proven) as well as Kang and Park's conjecture to general level. Utilizing a modified shift inequality, Inagaki and Tamura have recently proven that the Kang and Park conjecture holds for level $3$ in all but finitely many cases. They further conjectured a stronger shift inequality which would imply a general level result for all but finitely many cases. Here, we prove their conjecture for large enough $n$, generalize the result for an arbitrary shift, and discuss the implications for Alder-type partition inequalities.

Motivated by work of Chan, Chan, and Liu, we obtain a new general theorem which produces Ramanujan-Sato series for $1/\pi$. We then use it to construct explicit examples related to non-compact arithmetic triangle groups, as classified by Takeuchi. Some of our examples are new, and some reproduce existing examples., Comment: 33 pages

Integer partitions have long been of interest to number theorists, perhaps most notably Ramanujan, and are related to many areas of mathematics including combinatorics, modular forms, representation theory, analysis, and mathematical physics. Here, we focus on partitions with gap conditions and partitions with parts coming from fixed residue classes. Let $\Delta_d^{(a,b)}(n) = q_d^{(a)}(n) - Q_d^{(b)}(n)$ where $q_d^{(a)}(n)$ counts the number of partitions of $n$ into parts with difference at least $d$ and size at least $a$, and $Q_d^{(b)}(n)$ counts the number of partitions into parts $\equiv \pm b \pmod{d + 3}$. In 1956, Alder conjectured that $\Delta_d^{(1,1)}(n) \geq 0$ for all positive $n$ and $d$. This conjecture was proved partially by Andrews in 1971, by Yee in 2008, and was fully resolved by Alfes, Jameson and Lemke Oliver in 2011. Alder's conjecture generalizes several well-known partition identities, including Euler's theorem that the number of partitions of $n$ into odd parts equals the number of partitions of $n$ into distinct parts, as well as the first of the famous Rogers-Ramanujan identities. In 2020, Kang and Park constructed an extension of Alder's conjecture which relates to the second Rogers-Ramanujan identity by considering $\Delta_d^{(a,b,-)}(n) = q_d^{(a)}(n) - Q_d^{(b,-)}(n)$ where $Q_d^{(b,-)}(n)$ counts the number of partitions into parts $\equiv \pm b \pmod{d + 3}$ excluding the $d+3-b$ part. Kang and Park conjectured that $\Delta_d^{(2,2,-)}(n)\geq 0$ for all $d\geq 1$ and $n\geq 0$, and proved this for $d = 2^r - 2$ and $n$ even. We prove Kang and Park's conjecture for all but finitely many $d$. Toward proving the remaining cases, we adapt work of Alfes, Jameson and Lemke Oliver to generate asymptotics for the related functions. Finally, we present a more generalized conjecture for higher $a=b$ and prove it for infinite classes of $n$ and $d$., Comment: 24 pages

In a seminal 2007 paper, Andrews introduced a class of combinatorial objects that generalize partitions called $k$-marked Durfee symbols. Multivariate rank generating functions for these objects have been shown by many to have interesting modularity properties at certain vectors of roots of unity. Motivated by recent studies of rank generating functions for strongly unimodal sequences, we apply methods of Andrews to define an analogous class of combinatorial objects called $k$-marked strongly unimodal symbols that generalize strongly unimodal sequences. We establish a multivariate rank generating function for these objects, which we study combinatorially. We conclude by discussing potential quantum modularity properties for this rank generating function at certain vectors of roots of unity., Comment: 10 pages

Two analogues of the crank function are defined for overpartitions -- the first residual crank and the second residual crank. This suggests an exploration of crank functions defined for overpartitions whose parts are divisible by an arbitrary $d$. We examine the positive moments of these crank functions while varying $d$ and prove some inequalities.

Utilizing a classification due to Lemke Oliver of eta-quotients which are also theta functions (here called eta-theta functions), Folsom, Garthwaite, Kang, Treneer, and the fourth author constructed a catalog of mock modular forms $V_{mn}$ having weight $3/2$ eta-theta function shadows and showed that these mock modular forms when viewed on certain sets of rationals, transform as quantum modular forms under the action of explicit subgroups. In this paper, we introduce an infinite class of functions that generalizes one row of the catalog, namely the $V_{m1}$, and show that the functions in this infinite class are both mock modular and quantum modular forms., Comment: 16 pages

In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,\cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane $\mathbb H$, in work of Bringmann, Folsom, Garvan, Kimport, Mahlburg, and Ono. Quantum modular forms, defined by Zagier in 2010, are similar to modular or mock modular forms but are defined on the rationals $\mathbb Q$ as opposed to $\mathbb H$, and exhibit modular transformations there up to suitably analytic error functions in $\mathbb R$; in general, they have been related to diverse areas including number theory, topology, and representation theory. Here, we establish quantum modular properties of $R_n$., Comment: 24 pages. arXiv admin note: text overlap with arXiv:1810.05685

From the Modularity Theorem proven by Wiles, Taylor, et al, we know that all elliptic curves are modular. It has been shown by Martin and Ono exactly which are represented by eta-quotients, and some examples of elliptic curves represented by modular forms that are linear combinations of eta-quotients have been given by Pathakjee, RosnBrick, and Yoong. In this paper, we first show that eta-quotients which are modular for any congruence subgroup of level $N$ coprime to $6$ can be viewed as modular for $\Gamma_0(N)$. We then categorize when even weight eta-quotients can exist in $M_k (\Gamma_1(p))$ and $M_k (\Gamma_1(pq))$, for distinct primes $p,q$. We conclude by providing some new examples of elliptic curves whose corresponding modular forms can be written as a linear combination of eta-quotients, and describe an algorithmic method for finding additional examples., Comment: 19 pages

Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of $\rm{SL}_2(\mathbb Z)$, up to nontrivial error terms; however, their domains (the upper half-plane $\mathbb H$, and the rationals $\mathbb Q$, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more. In this paper we study the $(n+1)$-variable combinatorial rank generating function $R_n(x_1,x_2,\dots,x_n;q)$ for $n$-marked Durfee symbols. These are $n+1$ dimensional multisums for $n>1$, and specialize to the ordinary two-variable partition rank generating function when $n=1$. The mock modular properties of $R_n$ when viewed as a function of $\tau\in\mathbb H$, with $q=e^{2\pi i \tau}$, for various $n$ and fixed parameters $x_1, x_2, \cdots, x_n$, have been studied in a series of papers. Namely, by Bringmann and Ono when $n=1$ and $x_1$ a root of unity; by Bringmann when $n=2$ and $x_1=x_2=1$; by Bringmann, Garvan, and Mahlburg for $n\geq 2$ and $x_1=x_2=\dots=x_n=1$; and by the first and third authors for $n\geq 2$ and the $x_j$ suitable roots of unity ($1\leq j \leq n$). The quantum modular properties of $R_1$ readily follow from existing results. Here, we focus our attention on the case $n\geq 2$, and prove for any $n\geq 2$ that the combinatorial generating function $R_n$ is a quantum modular form when viewed as a function of $x \in \mathbb Q$, where $q=e^{2\pi i x}$, and the $x_j$ are suitable distinct roots of unity., Comment: 18 pages

We investigate the modular properties of a new partition rank, the $M_d$-rank of overpartitions. In fact this is an infinite family of ranks, indexed by the positive integer $d$, that gives both the Dyson rank of overpartitions and the overpartition $M_2$-rank as special cases. The $M_d$-rank of overpartitions is the holomorphic part of a certain harmonic Maass form of weight $\frac{1}{2}$. We give the exact transformation of this harmonic Maass form along with a few identities for the $M_d$-rank., Comment: 37 pages

We define a finite-field version of Appell-Lauricella hypergeometric functions built from period functions in several variables, paralleling the development by Fuselier, et. al in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite-field Appell-Lauricella functions to establish a finite-field analogue of Koike and Shiga's cubic transformation for the Appell hypergeometric function $F_1$, proving a conjecture of Ling Long. We use our multivariable period functions to construct formulas for the number of $\mathbb{F}_p$-points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite-field Appell-Lauricella functions., Comment: 20 pages; corrected typos

In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta-theta functions by constructing mock modular forms from the eta-theta functions with even characters, such that the shadows of these mock modular forms are given by the eta-theta functions with odd characters. In addition, we prove that our mock modular forms are quantum modular forms. As corollaries, we establish simple finite hypergeometric expressions which may be used to evaluate Eichler integrals of the odd eta-theta functions, as well as some curious algebraic identities., Comment: 33 pages

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without need for Hecke operators. In this paper we give a method for generalizing Lachterman, Schayer, and Younger's proof to include Ramanujan congruences for multipartition functions $p_k(n)$, and Ramanujan congruences for $p(n)$ modulo certain prime powers., Comment: 16 pages

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields and consequently study hypergeometric functions over finite fields in a manner that is parallel to that of the classical hypergeometric functions. Using a comparison between the classical gamma function and its finite field analogue the Gauss sum, we give a systematic way to obtain certain types of hypergeometric transformation and evaluation formulas over finite fields and interpret them geometrically using a Galois representation perspective. As an application, we obtain a few finite field analogues of algebraic hypergeometric identities, quadratic and higher transformation formulas, and evaluation formulas. We further apply these finite field formulas to compute the number of rational points of certain hypergeometric varieties., Comment: 113 pages

Based on work of Atkin and Swinnerton-Dyer on partition rank difference functions, and more recent work of Lovejoy and Osburn, Mao has proved several inequalities between partition ranks modulo $10$, and additional results modulo $6$ and $10$ for the $M_2$ rank of partitions without repeated odd parts. Mao conjectured some additional inequalities. We prove some of Mao's rank inequality conjectures for both the rank and the $M_2$ rank modulo $10$ using elementary methods., Comment: 12 pages

Ramanujan's celebrated congruences of the partition function $p(n)$ have inspired a vast amount of results on various partition functions. Kwong's work on periodicity of rational polynomial functions yields a general theorem used to establish congruences for restricted plane partitions. This theorem provides a novel proof of several classical congruences and establishes two new congruences. We additionally prove several new congruences which do not fit the scope of the theorem, using only elementary techniques, or a relationship to existing multipartition congruences., Comment: 9 pages, 2 figures

In 1997, van Hamme developed $p-$adic analogs, for primes $p$, of several series which relate hypergeometric series to values of the gamma function, originally studied by Ramanujan. These analogs relate truncated sums of hypergeometric series to values of the $p-$adic gamma function, and are called Ramanujan type supercongruences. In all, van Hamme conjectured 13 such formulas, three of which were proved by van Hamme himself, and five others have been proved recently using a wide range of methods. Here, we explore four of the remaining five van Hamme supercongruences, revisit some of the proved ones, and provide some extensions.

In this paper, we investigate the relationships among hypergeometric series, truncated hypergeometric series, and Gaussian hypergeometric functions through some families of `hypergeometric' algebraic varieties that are higher dimensional analogues of Legendre curves., Comment: 25 pages

This paper is devoted to abelian varieties arising from generalized Legendre curves. In particular, we consider their corresponding Galois representations, periods, and endomorphism algebras. For certain one parameter families of 2-dimensional abelian varieties of this kind, we determine when the endomorphism algebra of each fiber defined over the algebraic closure of $\Bbb Q$ contains a quaternion algebra.

We prove a conjecture due to Kimoto and Wakayama from 2006 concerning Apery-like numbers associated to a special value of a spectral zeta function. Our proof uses hypergeometric series and p-adic analysis., Comment: 8 pages, to appear in Proceedings of the AMS

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Gamma(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allow us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series., Comment: 14 pages, 1 figure