Supercongruences for sums involving fourth power of some rising factorials
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Supercongruences for sums involving fourth power of some rising factorials ARIJIT JANA and GAUTAM KALITA∗ Department of Science and Mathematics, Indian Institute of Information Technology Guwahati, Bongora 781 015, India *Corresponding author E-mail: [email protected]; [email protected]
MS received 2 August 2019; revised 20 December 2019; accepted 11 January 2020 Abstract. In this paper, we give proof of certain recent conjectural supercongruences on sums involving fourth power of some rising factorials. Keywords.
Supercongruences; hypergeometric series; rising factorial.
Mathematics Subject Classification.
11A07, 11D88, 33B15, 33C20.
1. Introduction and statement of results Following [1], we recall the definition of the gamma function given by ∞ (z) = t z−1 e−t dt, Re(z) > 0. 0
It satisfies the functional equation (1 + z) = z(z), which gives the continuation of (z) to a meromorphic function defined for all complex numbers z. Consequently, one can deduce that (a)n =
(a + n) , (a)
(1.1)
where the Pochhammer symbol or the rising factorial (a)k is defined as (a)0 := 1
(a)k := a(a + 1) · · · (a + k − 1) for k ≥ 1.
and
Throughout the paper, let p be an odd prime. For n ∈ N, the p-adic gamma function is defined as p (n) = (−1)n j. 0< j 3 by using hypergeometric series identities and evaluation. In [15], Long gave a generalization of (1.4) pr −1
2
k=0
( 1 )k (4k + 1) 2 k!
4 ≡ pr (mod pr +3 ),
which is recently proved by Guo and Wang [8] by establishing a q-analogue, and by Kalita and Jana [12] using hypergeometric series identities and evaluations. In this context, we prove a supercongruence related to (1.4) confirming a recent conjecture of Guo [3, Conjecture 4.3] for m = 3. Theorem 1.1. For any prime p > 3 and positive integer r , we have
pr −1
2
k=0
(4k + 1)
3
( 21 )k k!
4 ≡ − pr (mod pr +3 ).
It is to be noted here that Wang [21] proved Theorem 1.1 for r = 1 and p ≡ 2 (mod 3) using the powerful Zeilberger’s algorithm [16], and later Liu [13] proved Theorem 1.1 for
Proc. Indian Acad. Sci. (Math. Sci.)
(2020) 130:59
Page 3 of 13
59
r = 1 and all odd primes p. Moreover, Guo [4] proved the modulus pr +2 case of Theorem 1.1 by the method of creative microscoping introduced in [9] (see also [7]). Very recently, Hou et al. [10] reproved Theorem 1.1 for r = 1 among other things. Extending Theorem 1.1 to all integers > 2, we further prove the following supercongruence. Theorem 1.2. Let > 2 be a positive integer and p ≥ 5 a prime number such that p ≡ −1 (mod ). If r is a positive integer, then t pr −1
(2k + 1)3
k=0
( 1 )4k k!4
⎧ r 3r ⎪ (−1) 2 +1 p 2 (mod pr +3 ), if r is even and t = 1; ⎪ ⎨ p+1 r +1 3r +1 (−1)(2−) + p 2 ≡ (−1) 2 2 ⎪ (1− ) ⎪ p r +3 ), ⎩ (mod p if r is odd and t = − 1. 1 2 p (1− )
Recently, Guo and Schlosser [6] proved some supercongruences related to [20, (C.2)] including pr +1
2
(4k − 1)
k=0
4 ( −1 2 )k ≡ 0 (mod pr +3 ) k!4
(1.5)
as q-analog, by using q-Zeilberger algorithm. In this paper, we prove a gene
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