Related papers: On certain $q$-multiple sums
Kanade explored the construction of modular companions to $q$-series identities, using the asymptotics of Nahm sums, and Mizuno [Ramanujan J.\ {\bf 66} (2025), Paper No.\ 62, 31] recently obtained a generalization of Kanade's asymptotic…
The question of the convergence of generalized formal power series (with complex power exponents) solutions of $q$-difference equations is studied in the situation where the small divisors phenomenon arises; a sufficient condition of…
By applying the partial derivative operator to several summation formulas for hypergeometric series, we prove several double series for $\pi$ in this paper. Similarly, we also establish several $q$-analogues of them.
According to the method of series rearrangement, we establish the extensions of two $q$-Gosper identities with an extra integer parameter. The limiting cases of them produce the generalizations of Gosper's two $_3F_2(\frac{3}{4})$-series…
We give an $n$-space generalized $q$-binomial theorem, and some new $q$ series identities that resemble the traditional $q$ series partition generating functions. These identities enumerate stepping stone weighted vector partitions.
We construct new continued fraction expansions of Jacobi-type J-fractions in $z$ whose power series expansions generate the ratio of the $q$-Pochhamer symbols, $(a; q)_n / (b; q)_n$, for all integers $n \geq 0$ and where $a,b,q \in…
The duality is a fundamental property of the finite multiple harmonic sums (MHS). In this paper, we prove a duality result for certain generalizations of MHS which appear naturally as the differences of MHS. We also prove a formula for the…
According to the $q$-series method, a short proof for Hou and Sun's identity, which is the $q$-analogue of a known $\pi$-formula, is offered. Furthermore, $q$-analogues of several other $\pi$-formulas are also established in terms of the…
We prove two multivariate $q$-binomial identities conjectured by Bousseau, Brini and van Garrel [Geom. Topol. 28 (2024), 393-496, arXiv:2011.08830] which give generating series for Gromov-Witten invariants of two specific log Calabi-Yau…
We establish new product identities involving the $q$-analogue of the Fibonacci numbers. We show that the identities lead to alternate expressions of generating functions for close-packed dimers on non-orientable surfaces.
Based on a variant of Sury's polynomial identity we derive new expressions for various finite Fibonacci (Lucas) sums. We extend the results to Fibonacci and Chebyshev polynomials, and also to Horadam sequences. In addition to deriving sum…
We investigate a relationship between MacMahon's generalized sum-of-divisors functions and Chebyshev polynomials of the first kind. This determines a recurrence relation to compute these functions, as well as proving a conjecture of…
Recently, the concept of a D-analogue was introduced by the author. This is a Dirichlet series analogue for the already known and well researched hypergeometric q-series. we consider the D-analogues of the q-binomial coefficients, and a…
In the present paper, we investigate special generalized q-Euler numbers and polynomials. Some earlier results of T. Kim in terms of q-Euler polynomials with weight alpha can be deduced. For presentation of our formulas we apply the method…
In this note, we find a combinatorial identity which is closely related to the multi-dimensional integral $\gamma_{m}$ in the study of divisor functions. As an application, we determine the finite dual of the group algebra of infinite…
We prove new exact formulas for the generalized sum-of-divisors functions, $\sigma_{\alpha}(x) := \sum_{d|x} d^{\alpha}$. The formulas for $\sigma_{\alpha}(x)$ when $\alpha \in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over…
We give a $q$-analogue of some binomial coefficient identities of Y. Sun [Electron. J. Combin. 17 (2010), #N20] as follows: {align*} \sum_{k=0}^{\lfloor n/2\rfloor}{m+k\brack k}_{q^2}{m+1\brack n-2k}_{q} q^{n-2k\choose 2} &={m+n\brack…
By generalizing Gessel-Xin's Laurent series method for proving the Zeilberger-Bressoud $q$-Dyson Theorem, we establish a family of $q$-Dyson style constant term identities. These identities give explicit formulas for certain coefficients of…
A few years ago, the concept of a D-analogue was introduced as a Dirichlet series analogue for the already known and well researched hypergeometric q-series. The D-analogue of the q-Dixon sum is given here, in the context of seeing a direct…
We establish $q$-analogs for four congruences involving central binomial coefficients. The $q$-identities necessary for this purpose are shown via the $q$-WZ method.