Some q -Supercongruences from Transformation Formulas for Basic Hypergeometric Series
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Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series Victor J. W. Guo1 · Michael J. Schlosser2 Received: 29 March 2019 / Revised: 15 November 2019 / Accepted: 31 July 2020 © The Author(s) 2020
Abstract 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 qanalogues 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 φ11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new 12 φ11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials. Keywords Basic hypergeometric series · Supercongruences · Identities · Linearization
Communicated by Mourad Ismail. The first author was partially supported by the National Natural Science Foundation of China (Grant 11771175). The second author was partially supported by Austrian Science Fund Grant P32305.
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Michael J. Schlosser [email protected] Victor J. W. Guo [email protected]
1
School of Mathematics and Statistics, Huaiyin Normal University, Huai’an 223300, Jiangsu, People’s Republic of China
2
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
123
Constructive Approximation
Mathematics Subject Classification Primary 33D15; Secondary 11A07 · 11F33 · 33D45
1 Introduction Ramanujan, in his second letter to Hardy on February 27, 1913, mentioned the following identity ∞ ( 1 )5 2 (−1)k (4k + 1) 2 5k = , k! ( 43 )4 k=0
(1.1)
where (x) is the Gamma function and where (a)k = a(a + 1) · · · (a + k − 1) is the Pochhammer symbol. A p-adic analogue of (1.1) was conjectured by Van Hamme [55, Eq. (A.2)] as follows: ( p−1)/2
(−1) (4k k
k=0
( 1 )5 + 1) 2 5k k!
⎧ ⎨−
p
(mod p 3 ), if p ≡ 1 (mod 4), 3 4 ( ) p 4 ≡ ⎩ 0 (mod p 3 ), if p ≡ 3 (mod 4). (1.2)
Here and throughout the paper, p always denotes an odd prime and p (x) is the p-adic Gamma function. The congruence (1.2) was later proved by McCarthy and Osburn [43] through a combination of ordinary and Gaussian hypergeometric series. Recently, the congruence (1.2) for p ≡ 3 (mod 4) and p > 3 was further generalized by Liu [37] to the modulus p 4 case. It is well known that some truncated hypergeometric series are closely related to Calabi-
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