Related papers: Extensions of two $q$-Gosper identities with an ex…
In this note, we use a basic identity, derived from the generalized doubling integrals of \cite{C-F-G-K1}, in order to explain the existence of various global Rankin-Selberg integrals for certain $L$-functions. To derive these global…
Motivated by the works of Andrews-Merca and Guo-Zeng, we establish some truncated identities of Gauss by using some summation formulas from the works of Zhi-Guo Liu. These give three new expansions for partial sums of Gauss' triangular…
In a series of letters to D.Stanton, R.W.Gosper presented many strange evaluations of hypergeometric series. Recently, we rediscovered one of the strange hypergeometric identities appearing in [Go]. In this paper, we prove this identity and…
We prove two single-parameter q-supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of q-binomial theorem and a…
We study an extension to Fourier transforms of the old problem on absolute convergence of the re-expansion in the sine (cosine) Fourier series of an absolutely convergent cosine (sine) Fourier series. The results are obtained by revealing…
For any homogeneous identity between $q$-minors, we provide an identity between $P,Q$-minors.
In the present paper combinatorial identities involving q-dual sequences or polynomials with coefficients q-dual sequences are derived. Further, combinatorial identities for q-binomial coefficients(Gaussian coefficients), q-Stirling numbers…
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…
We present a systematic method for proving nonterminating basic hypergeometric identities. Assume that $k$ is the summation index. By setting a parameter $x$ to $xq^n$, we may find a recurrence relation of the summation by using the…
Foster's network theorems and their extensions to higher orders involve resistance values and conductances. We establish identities concerning voltage values and conductances. Our identities are analogous to the extended Foster's…
By making use of the multiplicate form of the extended Carlitz inverse series relations, we establish two general `dual' theorems of Jackson's summation formula for well--poised $_8\phi_7$-series. Their duplicate forms under the partition…
We study certain connections between the quantum invariants of the torus knots T_{3,2^k} and some q-series identities. In particular, we obtain new generalizations of Slater's identities (83) and (86).
We generalize a terminating summation formula to a unilateral nonterminating, and further, a bilateral summation formula by a property of analytic functions. The unilateral one is proved to be a $q$-analogue of a $_4F_3$-summation formula.…
We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.
The well-known Kummer's formula evaluates the hypergeometric series 2F1(A,B;C;-1) when the relation B-A+C=1 holds. This paper deals with evaluation of 2F1(-1) series in the case when C-A+B is an integer. Such a series is expressed as a sum…
It is shown that a wide range of probabilities and limiting probabilities in finite classical groups have integral coefficients when expanded as a power series in 1/q. Moreover it is proved that the coefficients of the limiting…
We give unified modular proofs to all of Gosper's identities on the $q$-constant $\Pi_q$. We also confirm Gosper's observation that for any distinct positive integers $n_1,\cdots,n_m$ with $m\geq 3$, $\Pi_{q^{n_1}}$, $\cdots$,…
Real count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial…
Recently, Feng, Kuznetsov and Yang discovered a very general reduction formula for a sum of products of the generalized hypergeometric functions (J. Math. Anal. Appl. 443(2016), 116--122). The main goal of this note is to present a…
Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…