## A common $$q$$-analogue of two supercongruences.(English)Zbl 1439.33007

This paper gives a $$q$$-congruences whose specializations $$q=1$$ and $$q= -1$$ correspond to supercongrences (B.2) and (H.2) on Van Hammer’s list in $$p$$-adic Functional A analysis. At the end a general common $$q$$-congruence for related hypergeometric sums is given.
This paper also, displays the following historical discussion, taking into account G. Bauer’s formula [J. Reine Angew. Math. 56, 101–121 (1859; ERAM 056.1478cj)],which traditionally gives different methods for proofs of hypergeometric identities, and its special status linked to the fact that it belongs to a family of series for $$\frac{1}{\pi}$$ of Ramanujan type and Ramanujan treatment and discussion in [S. Ramanujan, Quart. J. 45, 350–372 (1914; JFM 45.1249.01)].
Different methods used to prove this identity, some used hypergeometric functions, some used other methods such as a creative telescoping method which is a computer proof given by S. B. Ekhad and D. Zeilberger [in: Geometry, analysis and mechanics. Dedicated to Archimedes on his 2281st birthday. Singapore: World Scientific. 107–108 (1994; Zbl 0849.33003)] based on Wilf-Zilberger’s of creative telescoping. The paper also gives some historic links between congruences and their methods of proof.
The authors introduced and executed a new method of creative micro scoping to prove (and reprove) many $$q$$-analogues of classical supercongruences and on $$q$$-congruencies’ goal of this paper is to present new $$q$$-analogue of Van Hammer’s supercongruence, which was given in two theorems.
In section two a family of one parameter $$q$$-congruencies was presented through two theorems and one Lemma. A full discussion including some limiting cases is also presented. At the end one finds a list of references.

### MSC:

 33D15 Basic hypergeometric functions in one variable, $${}_r\phi_s$$ 11A07 Congruences; primitive roots; residue systems 11B65 Binomial coefficients; factorials; $$q$$-identities

### Citations:

Zbl 0849.33003; ERAM 056.1478cj; JFM 45.1249.01
Full Text:

### References:

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