On two congruences involving Franel numbers
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On two congruences involving Franel numbers Ji-Cai Liu1 Received: 10 April 2020 / Accepted: 9 September 2020 / Published online: 13 September 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract Via symbolic summation method, we establish the following series for π 2 : ∞ π2 Hk − 2H2k = , (−3)k k 18 k=1
k
where Hk = j=1 1/ j. We also derive a p-adic congruence related to this series. As an application, we prove two congruences involving Franel numbers, one of which was originally conjectured by Sun. Keywords Congruences · Harmonic numbers · Franel numbers · Bernoulli polynomials Mathematics Subject Classification 33F10 · 11A07 · 05A19
1 Introduction It is well-known that (see [33, (3)]) ∞ k=1
π2 1 , 2k = 18 k2
(1.1)
k
which can be derived from the following familiar power series expansion: 2 (arcsin(x/2))2 =
∞ x 2k . 2 2k k=1 k k
Recall that the Euler numbers and Bernoulli polynomials are defined as ∞
2 xn = En x −x e +e n! n=0
B 1
Ji-Cai Liu [email protected] Department of Mathematics, Wenzhou University, Wenzhou 325035, People’s Republic of China
123
201
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J.-C. Liu
and ∞
xet x xk B (t) = . k ex − 1 k! k=0
In 2011, Sun [26, (1.3)] showed that (1.1) possesses the following interesting p-adic analogue: ( p−1)/2 k=1
1
k 2 2k k
4 ≡ (−1)( p−1)/2 E p−3 3
(mod p),
for any prime p ≥ 5. Mattarei and Tauraso [21, Theorem 6.1] investigated congruence −1 k p−1 properties for the polynomial k=1 k −2 2k t . k In recent decades, infinite sums on harmonic numbers related to powers of π have been widely studied. For instance, Borwein and J.M. Borwein [1] proved that ∞ Hk2 17 4 = π , 2 k 360 k=1
where the kth generalized harmonic number is given by (r )
Hk
=
k 1 , jr j=1
(1) Hk .
It is worth mentioning that Sun [27,31] conjectured with the convention that Hk = and proved many infinite identities involving harmonic numbers related to powers of π. For more series for π and related congruences as well as q-congruences, one can refer to [3,4,7–10,14–19,26,28,34]. The first aim of the paper is to establish the following series for π 2 . Theorem 1.1 We have ∞ Hk − 2H2k π2 = . (−3)k k 18
(1.2)
k=1
Our proof of (1.2) makes use of a finite identity, which is derived from symbolic summation method. We also show that (1.2) has the following p-adic analogue. Theorem 1.2 For any prime p ≥ 5, we have ( p−1)/2
where
· 3
k=1
Hk − 2H2k 1 p ≡ B p−2 (−3)k k 6 3
1 3
(mod p),
denotes the Legendre symbol.
In 1894, Franel [5] found that the sums of cubes of binomial coefficients: n 3 n fn = k k=0
satisfy the recurrence: (n + 1)2 f n+1 = (7n 2 + 7n + 2) f n + 8n 2 f n−1 .
123
(1.3)
On two congruences involving Franel numbers
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The numbers f n are known as Franel numbers, which also appear in Strehl’s identity (see [25]): n 2 n n 2k n k 2k fn = = . (1.4) k n k n−k k k=0
k=0
Since the appearance of the Franel numbers, some interesting congruence properties have been gradually discovered (see [6,12,29]). For inst
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