Electronic Research Archive
Special issue on combinatorial number Th. & enumerative combinatorics
Zhi-Wei Sun, Department of Mathematics, Nanjing University, Nanjing 210093, China email@example.com
Jiang Zeng, Institut Camille JordanUniversité Claude Bernard Lyon 143, boulevard du 11 novembre 1918F-69622 Villeurbanne Cedex, France firstname.lastname@example.org
The pentagonal numbers are the integers given by
We confirm a conjectural supercongruence involving Catalan numbers, which is one of the 100 selected open conjectures on congruences of Sun. The proof makes use of hypergeometric series identities and symbolic summation method.
Using Watson's terminating
Via symbolic computation we deduce 97 new type series for powers of
where the generalized central trinomial coefficient
Eighteen of the new series in this paper involve some imaginary quadratic fields with class number
Flajolet and Françon [European. J. Combin. 10 (1989) 235-241] gave a combinatorial interpretation for the Taylor coefficients of the Jacobian elliptic functions in terms of doubled permutations. We show that a multivariable counting of the doubled permutations has also an explicit continued fraction expansion generalizing the continued fraction expansions of Rogers and Stieltjes. The second goal of this paper is to study the expansion of the Taylor coefficients of the generalized Jacobian elliptic functions, which implies the symmetric and unimodal property of the Taylor coefficients of the generalized Jacobian elliptic functions. The main tools are the combinatorial theory of continued fractions due to Flajolet and bijections due to Françon-Viennot, Foata-Zeilberger and Clarke-Steingrímsson-Zeng.
The Euler number
Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.
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