advertisement

BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES DERMOT McCARTHY Abstract. We establish two binomial coefficient–generalized harmonic sum identities using the partial fraction decomposition method. These identities are a key ingredient in the proofs of numerous supercongruences. In particular, in other works of the author, they are used to establish modulo pk (k > 1) congruences between truncated generalized hypergeometric series, and a function which extends Greene’s hypergeometric function over finite fields to the p-adic setting. A specialization of one of these congruences is used to prove an outstanding conjecture of Rodriguez-Villegas which relates a truncated generalized hypergeometric series to the p-th Fourier coefficient of a particular modular form. 1. Introduction and Statement of Results (i) For non-negative integers i and n, we define the generalized harmonic sum, Hn , by n X 1 (i) Hn := ji j=1 (i) and H0 := 0. In [3] Chu proves the following binomial coefficient-generalized harmonic sum identity using the partial fraction decomposition method. If n is a positive integer, then n X n+k 2 n 2 (1) (1) (1) 1 + 2kHn+k + 2kHn−k − 4kHk = 0. (1.1) k k k=1 This identity had previously been established using the WZ method [1] and was used by Ahlgren and Ono in proving the Apéry number supercongruence [2]. In [4], [5] the author establishes various supercongruences between truncated generalized hypergeometric series, and a function which extends Greene’s hypergeometric function over finite fields to the p-adic setting. Specifically, let p be an odd prime and let n ∈ Z+ . For mi i 1 ≤ i ≤ n + 1, let m di ∈ Q ∩ Zp such that 0 < di < 1. Let Γp (·) denote Morita’s p-adic gamma function, bxc denote the greatest integer less than or equal to x and hxi denote the fractional part of x, i.e. x − bxc. Then define mn+1 m1 m2 n+1 G d1 , d2 , . . . , dn+1 p j p−2 i n+1 n+1 m Y Γp h m −1 X −b i − j c di − p−1 i j j := (−1) Γp p−1 (−p) di p−1 . mi p−1 Γp di j=0 i=1 2010 Mathematics Subject Classification. Primary: 05A19; Secondary: 11B65. This work was supported by the UCD Ad Astra Research Scholarship program. 1 2 DERMOT McCARTHY Note that when p ≡ 1 (mod di ) this function recovers Greene’s hypergeometric function over finite fields. For a complex number a and a non-negative integer n let (a)n denote the rising factorial defined by (a)0 := 1 and (a)n := a(a + 1)(a + 2) · · · (a + n − 1) for n > 0. Then, for complex numbers ai , bj and z, with none of the bj being negative integers or zero, we define the truncated generalized hypergeometric series " r Fs a1 , a2 , a3 , . . . , ar z b1 , b2 , . . . , bs # := m X (a1 )n (a2 )n (a3 )n · · · (ar )n z n . (b1 )n (b2 )n · · · (bs )n n! n=0 m An example of one the supercongruence results from [5] is the following theorem. Theorem 1.1 ([5] Thm. 2.7). Let r, d ∈ Z such that 2 ≤ r ≤ d − 2 and gcd(r, d) = 1. Let 2 p be an odd primesuch that ≡ ±1 p d−1 (mod d) or p ≡ ±r (mod d) with r ≡ ±1 (mod d). If 1 r d−r s(p) := Γp d Γp d Γp d Γp d , then 4G 1 r d, d, 1 − r d, 1 − 1 d p " ≡ 4 F3 r d, 1 d, 1, 1 − dr , 1 − 1, 1 1 d # 1 + s(p)p (mod p3 ). p−1 A specialization of this congruence is used to prove an outstanding supercongruence conjecture of Rodriguez-Villegas, which relates a truncated generalized hypergeometric series to the p-th Fourier coefficient of a particular modular form [4],[6]. Similar results to Theorem 1.1 exist for 4 G with other parameters, and also 2 G and 3 G. The main results of the current paper, Theorems 1.2 and 1.3 below, are two binomial coefficient– generalized harmonic sum identities which factor heavily into the proofs of all the 4 G congruences. Taking particular values for n, m, l, c1 and c2 in these identities allows the vanishing of certain terms in the proofs. Note that letting m = n in Theorem 1.2 recovers (1.1). Theorem 1.2. Let m, n be positive integers with m ≥ n. Then n X m+k m n+k n (1) (1) (1) (1) (1) 1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk k k k k k=0 + m X k−n (−1) k=n+1 m+k k m k n+k . k−1 = (−1)m+n . k n BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES 3 Theorem 1.3. Let l, m, n be positive integers with l > m ≥ n ≥ 2l and c1 , c2 ∈ Q some constants. Then ( n X m+k m n+k n (1) (1) (1) (1) (1) 1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk k k k k k=0 · c1 + c2 (1) Hk+n (2) Hk+m − (1) Hk+l−n−1 − (2) Hk+l−m−1 + (1) c2 Hk+m ) + m X − (1) Hk+l−m−1 k−n (−1) k=n+1 (2) (2) − k c1 Hk+n − Hk+l−n−1 m+k k m k n+k . k−1 k n (1) (1) (1) (1) · c1 Hk+n − Hk+l−n−1 + c2 Hk+m − Hk+l−m−1 = 0. The remainder of this paper is spent proving Theorems 1.2 and 1.3. 2. Proofs We first develop two algebraic identities of which the binomial coefficient–harmonic sum identities are limiting cases. Theorem 2.1. Let x be an indeterminate and let m, n positive integers with m ≥ n. Then n X m+k m n+k n k=0 k k ( · + k −k + (x + k)2 k (1) (1) (1) (1) (1) ) 1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk x+k m X x(1 − x)n (1 − x)m (−1)k−n m + k m n+k . k−1 = . x+k k k k n (x)n+1 (x)m+1 (2.1) k=n+1 Proof. Using partial fraction decomposition we can write n m X x(1 − x)n (1 − x)m A X Bk Ck Dk f (x) := = + + + (x)n+1 (x)m+1 x (x + k)2 x + k x+k k=1 k=n+1 for some A, Bk , Ck and Dk ∈ Q. We now isolate these coefficients by taking various limits of f (x) as follows. A = lim xf (x) = lim x→0 x→0 (1 − x)n (1 − x)m = 1. (1 + x)n (1 + x)m 4 DERMOT McCARTHY For 1 ≤ k ≤ n, x(1 − x)n (1 − x)m Bk = lim (x + k)2 f (x) = lim x→−k (x)2 (x k x→−k = = + k + 1)n−k (x + k + 1)m−k −k(k + 1)n (k + 1)m (−k)2k (1)n−k (1)m−k −k(k + 1)n (k + 1)m (−1)2k k!2 (n − k)!(m − k)! m+k = −k k m k n+k k n , k and, using L’Hôspital’s rule, (x + k)2 f (x) − Bk x→−k x+k Ck = lim " # d = lim (x + k)2 f (x) x→−k dx " # x(1 − x)n (1 − x)m d = lim x→−k dx (x)2 (x + k + 1) k n−k (x + k + 1)m−k (" = lim x→−k + m X (x)2k (x + k + 1)n−k (x + k + 1)m−k −1 (−x + s) + s=1 " = #" (1 − x)n (1 − x)m n−k X −1 (x + k + s) s=1 (1 + k)n (1 + k)m (−k)2k (1)n−k (1)m−k #" 1+k + m−k X 1−x −1 (x + k + s) = m+k k m k +2 s=1 n X s=1 (k + s)−1 + k−1 X !#) −1 (x + s) s=0 m X (k + s)−1 + s=1 m−k X s=1 (−x + s)−1 s=1 + n X n−k X (s)−1 s=1 k−1 X (s)−1 + 2 !# (−k + s)−1 s=0 n+k n (1) (1) (1) (1) (1) 1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk . k k BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES 5 Similarly, for n + 1 ≤ k ≤ m, x(1 − x)n (1 − x)m x→−k (x)n+1 (x)k (x + k + 1)m−k Dk = lim (x + k)f (x) = lim x→−k = −k(k + 1)n (k + 1)m (−k)n+1 (−k)k (1)m−k = (−1)k−n m+k k m k n+k . k−1 . k n Theorem 2.2. Let x be an indeterminate and let l, m, n be positive integers with l > m ≥ n ≥ and c1 , c2 ∈ Q some constants. Then ( n X 1 m+k m n+k n (1) (1) c1 Hk+n − Hk+l−n−1 x+k k k k k k=0 x (1) (1) (1) (1) (1) (1) (1) + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk + c2 Hk+m − Hk+l−m−1 · x+k ) (2) (2) (2) (2) − k c1 Hk+n − Hk+l−n−1 + c2 Hk+m − Hk+l−m−1 + l 2 m X (−1)k−n m + k m n+k . k−1 x+k k k k n k=n+1 · c1 (1) Hk+n − (1) Hk+l−n−1 + c2 (1) Hk+m − (1) Hk+l−m−1 " # n m X X x(1 − x)n (1 − x)m = c1 (−x + s)−1 + c2 (−x + s)−1 . (2.2) (x)n+1 (x)m+1 s=l−n s=l−m Proof. Using partial fraction decomposition we can write " # n m X X x(1 − x)n (1 − x)m c1 (−x + s)−1 + c2 (−x + s)−1 f (x) : = (x)n+1 (x)m+1 s=l−n s=l−m n m X A X Bk Ck Dk = + + + 2 x (x + k) x+k x+k k=1 k=n+1 for some A, Bk , Ck and Dk ∈ Q. As in the proof of Theorem 2.1, we isolate the coefficients A, Bk , Ck and Dk by taking various limits of f (x). For brevity, we first let 6 DERMOT McCARTHY n X Ta(r) := c1 (a + s)−r + c2 s=l−n m X (a + s)−r s=l−m and (r) (r) (r) (r) U (r) := c1 Hk+n − Hk+l−n−1 + c2 Hk+m − Hk+l−m−1 . Then we have n X A = lim xf (x) = c1 lim x→0 x→0 s=l−n n X = c1 m X (1 − x)n (1 − x)m (1 − x)n (1 − x)m + c2 lim x→0 (1 + x)n (1 + x)m (s − x) (1 + x)n (1 + x)m (s − x) s=l−m m X s−1 + c2 s=l−n s−1 s=l−m (1) (1) (1) = c1 Hn(1) − Hl−n−1 + c2 Hm − Hl−m−1 . For 1 ≤ k ≤ n, x(1 − x)n (1 − x)m Bk = lim (x + k)2 f (x) = lim x→−k (x)2 (x k x→−k = (1) + k + 1)n−k (x + k + 1)m−k −k(k + 1)n (k + 1)m T−x (1) T (−k)2k (1)n−k (1)m−k k m+k m n+k n = −k U (1) k k k k and " # d 2 Ck = lim (x + k) f (x) x→−k dx " # x(1 − x)n (1 − x)m d (1) = lim T−x x→−k dx (x)2 (x + k + 1) k n−k (x + k + 1)m−k (" #" (1 − x)n (1 − x)m (2) (1) (1) = lim x T−x + T−x − x T−x 2 x→−k (x)k (x + k + 1)n−k (x + k + 1)m−k · n m n−k X X X −1 −1 (−x + s) + (−x + s) + (x + k + s)−1 s=1 s=1 s=1 + m−k X s=1 (x + k + s) −1 +2 k−1 X s=0 !#) (x + s) −1 BINOMIAL COEFFICIENT–HARMONIC SUM IDENTITIES ASSOCIATED TO SUPERCONGRUENCES " = (1 + k)n (1 + k)m #" (2) (1) −kTk + Tk (−k)2k (1)n−k (1)m−k 1+k + (s)−1 + s=1 = n m X X (k + s)−1 + (k + s)−1 s=1 n−k X m−k X 7 s=1 (s)−1 + 2 s=1 k−1 X !!# (−k + s)−1 s=0 m+k m n+k n k k k k (1) (1) (1) (1) (1) (2) (1) . · −k U + 1 + k Hm+k + Hm−k + Hn+k + Hn−k − 4Hk U For n + 1 ≤ k ≤ m, Dk = lim (x + k)f (x) = lim x→−k x→−k = x(1 − x)n (1 − x)m (1) T (x)n+1 (x)k (x + k + 1)m−k −x −k(k + 1)n (k + 1)m (1) T (−k)n+1 (−k)k (1)m−k k k−n = (−1) U (1) m+k k m k n+k . k−1 . k n Proofs of Theorems 1.2 and 1.3. Multiply both sides of (2.1) and (2.2) respectively by x and take the limit as x → ∞. References [1] S. Ahlgren, S. B. Ekhad, K. Ono, D. Zeilberger, A binomial coefficient identity associated to a conjecture of Beukers, Electron. J. Combin. 5 (1998), Research Paper 10, 1. [2] S. Ahlgren, K. Ono, A Gaussian hypergeometric series evaluation and Apéry number congruences, J. Reine Angew. Math. 518 (2000), 187–212. [3] W. Chu, A binomial coefficient identity associated with Beukers’ conjecture on Apéry numbers, Electron. J. Combin. 11 (2004), no. 1, Note 15, 3. [4] D. McCarthy, p-adic hypergeometric series and supercongruences, Ph.D thesis, University College Dublin, 2010. [5] D. McCarthy, Extending Gaussian hypergeometric series to the p-adic setting, Int. J. Number Theory, accepted for publication, 24 pages. [6] D. McCarthy, On a supercongruence conjecture of Rodriguez-Villegas, Proc. Amer. Math. Soc. 140 (2012), 2241–2254. Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA E-mail address: [email protected]