In this paper we introduce the so-called truncated very-well-poised $_6psi_6$ series and set up an explicit recurrence relation for it by means of the classical Abel lemma on summation by parts. This new recurrence relation implies an elementary proof of Baileys well-known $_6psi_6$ summation formula.
Andrews and Merca investigated a truncated version of Eulers pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobis triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Eulers pentagonal number theorem. Meanwhile, we find an analogous result for a truncated series arising from Jacobis triple product identity in a different manner.
We generalize Dahmen-Micchelli deconvolution formula for Box splines with parameters. Our proof is based on identities for Poisson summation of rational functions with poles on hyperplanes.
We introduce an L-series associated with harmonic Maass forms and prove their functional equations. We establish converse theorems for these L-series and, as an application, we formulate and prove a summation formula for the holomorphic part of a harmonic lift of a given cusp form.
A new recurrence relation for exceptional orthogonal polynomials is proposed, which holds for type 1, 2 and 3. As concrete examples, the recurrence relations are given for Xj-Hermite, Laguerre and Jacobi polynomials in j = 1,2 case.
Inspired by the recent work on $q$-congruences and the quadratic summation formula of Rahman, we provide some new $q$-supercongruences. By taking $qto 1$ in one of our results, we obtain a new Ramanujan-type supercongruence, which has the same right-hand side as Van Hammes (G.2) supercongruence for $pequiv 1 pmod 4$. We also formulate some related challenging conjectures on supercongruences and $q$-supercongruences.
Jin Wang
,Xinrong Ma
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(2021)
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"A new recurrence relation for the truncated very-well-poised $_6psi_6$ series and Baileys summation formula"
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Ma Xinrong
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