相关论文: An asymptotic formula of the divergent bilateral b…
We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1\psi_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection…
We give the new connection formula for the divergent bilateral basic hypergeometric series ${}_2\psi_2(a_1,a_2;b_1;q,x)$ by the using of the $q$-Borel-Laplace resummation method and Slater's formula. The connection coefficients are given by…
We study the divergent basic hypergeometric series which is a $q$-analog of divergent hypergeometric series. This series formally satisfies the linear $q$-difference equation. In this paper, for that equation, we give an actual solution…
We give summation formulae for the bilateral basic hypergeometric series ${}_1\psi_1( a; b; q, z )$ through Ramanujan's summation formula, which are generalizations of nontrivial identities found in the physics of three-dimensional Abelian…
We examine the convergence of $q$-hypergeometric series when $|q|=1$.
We show the connection formula for the basic hypergeometric series ${}_3\varphi_1(a_1,a_2,a_3;b_1;q,x)$ between around the origin and infinity by the using of the $q$-Borel-Laplace transformations. We also show the limit $q\to 1-0$ of the…
Usually when solving differential or difference equations via series solutions one encounters divergent series in which the coefficients grow like a factorial. Surprisingly, in the $q$-world the $n$th coefficient is often of the size…
We prove an asymptotic formula for a special case of the Gauss hypergeometric function which arises in explicit formulas for moments of Maass form symmetric square L-functions. The resulting formula is uniform in several variables, which is…
We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.
We show how to determine the asymptotics of a certain Selberg-type integral by means of tools available in the theory of (generalised) hypergeometric series. This provides an alternative derivation of a result of Carr\'e, Deneufch\^atel,…
We characterize an asymptotic mean value formula in the viscosity sense for the double phase elliptic equation $$ -{\rm div}(\lvert \nabla u \rvert^{p-2}\nabla u+ a(x)\lvert\nabla u \rvert^{q-2}\nabla u)=0 $$ and the normalized double phase…
We use a new $q$-exponential operator based on the $q^{\pm1}$-derivative $\D_{q^{\pm1}}$ of order 1 to derive summation formulas for bilateral basic hypergeometric series ${}_{0}\psi_{1}$, ${}_{1}\psi_{1}$, ${}_{1}\psi_{2}$, and…
This paper establishes new results concerning asymptotic expansions of $q$-series related to partial theta functions. We first establish a new method to obtain asymptotic expansions using a result of Ono and Lovejoy, and then build on these…
Let $\mu_1$ and $\mu_2$ be two complex-valued Borel measures on the real line such that $\operatorname{supp} \mu_1 =[\alpha_1,\beta_1] < \operatorname{supp} \mu_2 =[\alpha_2,\beta_2]$ and ${\rm d}\mu_i(x) = -\rho_i(x){\rm d}x/2\pi {\rm i}$,…
We consider the uniform asymptotic expansion for the Gauss hypergeometric function \[{}_2F_1(a+\epsilon\lambda,b;c+\lambda;x),\qquad 0<x<1\] as $\lambda\to+\infty$ in the neigbourhood of $\epsilon x=1$ when the parameter $\epsilon>1$ and…
The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose…
We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbf{\Sigma}^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated…
A new approach to summation of divergent field-theoretical series is suggested. It is based on the Borel transformation combined with a conformal mapping and does not imply the knowledge of the exact asymptotic parameters. The method is…
We prove some new semi-finite forms of bilateral basic hypergeometric series. One of them yields in a direct limit Bailey's celebrated ${}_6\psi_6$ summation formula, answering a question recently raised by Chen and Fu ({\em Semi-Finite…
We address the problem of ambiguity of a function determined by an asymptotic perturbation expansion. Using a modified form of the Watson lemma recently proved elsewhere, we discuss a large class of functions determined by the same…