Congruences on sums of q-binomial coefficients

Several new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

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“…We refer the reader to [1,7,[16][17][18][19][20][21][22][23][24]27,36,39,45,49,53,54,57,58] for some interesting q-congruences.…”

Section: Introductionmentioning

confidence: 99%

Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

Guo

Schlosser

2020

Constr Approx

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“…It is easy to see that lim q→1 C n (q) = 2n n /(n + 1) is the ordinary nth Catalan number. Recently, q-analogues of congruences (q-congruences) for indefinite sums of binomial coefficients as well as hypergeometric series have attracted attention (see, for example, [4,5,7,10,12]).…”

Section: It Follows Thatmentioning

confidence: 99%

A Truncated Identity of Euler and Related -Congruences

Liu

Huang

2020

Bull. Aust. Math. Soc.

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We discuss a truncated identity of Euler and present a combinatorial proof of it. We also derive two finite identities as corollaries. As an application, we establish two related $q$ -congruences for sums of $q$ -Catalan numbers, one of which has been proved by Tauraso [‘ $q$ -Analogs of some congruences involving Catalan numbers’, Adv. Appl. Math. 48 (2012), 603–614] by a different method.

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“…During the past few years, q-analogues of congruences and supercongruences have been investigated by many authors (see [3][4][5][6][7][8][9][10][11][12][13][14][15][16]18,21,22,24,25,27]). For instance, using a method similar to that used in [26], the first author and Wang [12,Theorem 1.2] gave a q-analogue of (1.2): for odd n,…”

Section: Introductionmentioning

confidence: 99%

“…During the past few years, q -analogues of congruences and supercongruences have been investigated by many authors (see [ 3 – 16 , 18 , 21 , 22 , 24 , 25 , 27 ]). For instance, using a method similar to that used in [ 26 ], the first author and Wang [ 12 , Theorem 1.2] gave a q -analogue of ( 1.2 ): for odd n , Moreover, the first author and Zudilin [ 13 ] devised a method of ‘creative microscoping’ to prove that, for any positive integer n with , where is the Jacobi symbol, see [ 13 , Theorem 1.1, Equation (6)].…”

Section: Introductionmentioning

confidence: 99%