arXiv subjects

Mohit Tripathi

Publications and source records attributed to Mohit Tripathi.

5 recordsLinked to original sources

arXiv2026

Sign Patterns in a Two Colored Partition Companion series

We study two closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series \[ S_1(q)=\sum_{n\ge0}s_1(n)q^n=\sum_{a\ge0}q^a(-q^{a+1};q)_\infty^2 \] and its odd companion, denoted by $T_o(q)$. First, for the eta-normalized companion \[ C(q)=(q;q)_\infty T_o(q)=\sum_{n\ge0}c(n)q^n, \] we prove a strong form of the Andrews--El Bachraoui sign conjecture that $\limsup c(n)=+\infty$ and $\liminf c(n)=-\infty$. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews--El Bachraoui congruence for $s_1(n)$ modulo 4.

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arXiv2026

Sparse Distribution of Coefficients of $\ell$-fold Product $L$-functions at Integers Represented by Quadratic Forms

Let $f \in S_{k}(\Gamma_{0}(N))$ be a normalized Hecke eigenform. We study the Fourier coefficients $\lambda_{f \otimes \cdots \otimes_{\ell} f}(n)$ of the $\ell$-fold product $L$-function for odd $\ell \ge 3$. Our focus is the distribution of this sequence over the sparse set of integers represented by a primitive, positive-definite binary quadratic form $Q$ of a fixed discriminant $D$. We establish an explicit upper bound for the summatory function of these coefficients, with dependencies on the weight, level, and discriminant. As a key application, we provide a bound for the first sign change of the sequence in this setting. We also generalize this result to find the first sign change among integers represented by any of the $h(D)$ forms of discriminant $D$, showing the bound improves as the class number increases.

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arXiv2024

Splitting Hypergeometric Functions over Roots of Unity

We examine hypergeometric functions in the finite field, p-adic and classical settings. In each setting, we prove a formula which splits the hypergeometric function into a sum of lower order functions whose arguments differ by roots of unity. We provide multiple applications of these results, including new reduction and summation formulas for finite field hypergeometric functions, along with classical analogues; evaluations of special values of these functions which apply in both the finite field and p-adic settings; and new relations to Fourier coefficients of modular forms.

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arXiv2020

Certain product formulas and values of Gaussian hypergeometric series

In this article we find finite field analogues of certain product formulas satisfied by the classical hypergeometric series. We express product of two ${_2}F_1$-Gaussian hypergeometric series as ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. We use properties of Gauss and Jacobi sums and our earlier works on finite field Appell series to deduce these product formulas satisfied by the Gaussian hypergeometric series. We then use these transformations to evaluate explicitly some special values of ${_4}F_3$- and ${_3}F_2$-Gaussian hypergeometric series. By counting points on CM elliptic curves over finite fields, Ono found certain special values of ${_2}F_1$- and ${_3}F_2$-Gaussian hypergeometric series containing trivial and quadratic characters as parameters. Later, Evans and Greene found special values of certain ${_3}F_2$-Gaussian hypergeometric series containing arbitrary characters as parameters from where some of the values obtained by Ono follow as special cases. We show that some of the results of Evans and Greene follow from our product formulas including a finite field analogue of the classical Clausen's identity.

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arXiv2018

A finite field analogue of the Appell series $F_4$

We define a function $F_4^{\ast}$ as a finite field analogue of the classical Appell series $F_4$ using Gauss sums. We establish identities for $F_4^{\ast}$ analogous to those satisfied by the classical Appell series $F_4$.

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