Zhi-Wei Sun Department of Mathematics, Nanjing University Nanjing , People s Republic of China

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1 Ramanuan J. 40(2016, no. 3, CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x Zhi-Wei Sun Deartment of Mathematics, Naning University Naning , Peole s Reublic of China zwsun@nu.edu.cn htt://math.nu.edu.cn/ zwsun Abstract. Define g n (x n ( n 2 ( 2 0 x for n 0, 1, 2,.... Those numbers g n g n (1 are closely related to Aéry numbers and Franel numbers. In this aer we establish some fundamental congruences involving g n (x. For examle, for any rime > 5 we have 1 1 g ( 1 0 (mod 2 and 1 1 g ( (mod. This is similar to Wolstenholme s classical congruences (mod 2 and (mod 2 for any rime > 3. It is well nown that n 0 ( 2 n 1. Introduction ( 2n n (n 0, 1, 2,... and central binomial coefficients lay imortant roles in mathematics. A famous theorem of J. Wolstenholme [W] asserts that for any rime > 3 we have ( ( (mod 3, Mathematics Subect Classification. Primary 11A07, 11B65; Secondary 05A10, 05A30, 11B75. Keywords. Franel numbers, Aéry numbers, binomial coefficients, congruences. Suorted by the National Natural Science Foundation (Grant No of China. 1

2 2 ZHI-WEI SUN where H n : H 1 0 (mod 2 and H ( < n 1 and H (2 n : 0< n (mod, 1 2 for n N {0, 1, 2,... }; see also [Zh] for some extensions. The reader may consult [S11a], [S11b], [ST1] and [ST2] for recent wor on congruences involving central binomial coefficients. The Franel numbers given by f n n 0 ( 3 n (n 0, 1, 2,... (cf. [Sl, A000172] were first introduced by J. Franel in 1895 who noted the recurrence relation: (n f n+1 (7n(n f n + 8n 2 f (n 1, 2, 3,.... In 1992 C. Strehl [St92] showed that the Aéry numbers given by A n n ( 2 ( 2 n n + 0 n ( 2 ( 2 n + 2 (n 0, 1, 2, (arising from Aéry s roof of the irrationality of ζ(3 n1 1/n3 (cf. [vp] can be exressed in terms of Franel numbers, namely, Define g n A n 0 n 0 ( n ( n + n ( 2 ( n 2 f. (1.1 for n N. (1.2 Such numbers are interesting due to Barrucand s identity ([B] n 0 ( n f g n (n 0, 1, 2,.... (1.3 For a combinatorial interretation of such numbers, see D. Callan [C]. The seuences (f n n 0 and (g n n 0 are two of the five soradic seuences (cf. D. Zagier [Z, Section 4] which are integral solutions of certain Aéry-lie recurrence euations and closely related to the theory of modular forms.

3 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 3 In [S12] and [S13b] the author introduced the Aéry olynomials A n (x : n ( 2 ( 2 n n + x (n 0, 1, 2,... 0 and the Franel olynomials f n (x : n 0 ( n 2 ( 2 n x n 0 ( ( n n ( 2 x (n 0, 1, 2,..., and deduced various congruences involving such olynomials. (Note that A n (1 A n, and f n (1 f n by [St94]. See also [S13a] for connections between rimes x 2 + 3y 2 and the Franel numbers. Here we introduce the olynomials g n (x : n 0 ( n 2 ( 2 x (n 0, 1, 2,.... Both f n (x and g n (x lay imortant roles in some inds of series for 1/π (cf. Conecture 3 and the subseuent remar in [S11]. In this aer we study various congruences involving g n (x. As usual, for an odd rime and an integer a, ( a denotes the Legendre symbol, and (a stands for the Fermat uotient (a 1 1/ if a. Also, B 0, B 1, B 2,... are the well-nown Bernoulli numbers and E 0, E 1, E 2,... are the Euler numbers. Now we state our main results. Theorem 1.1. Let > 3 be a rime. (i We have 0 Conseuently, g (x(1 2 H (2 g ( 1 g ( g ( ( H (2 x (mod 4. (1.4 g H ( B 3 (mod 4, (1.5 ( g ( 1H (2 E 3 (mod 3, (1.6 (mod 2. (1.7

4 4 ZHI-WEI SUN (ii We also have and moreover 1 1 g (x g 1 0 (mod, ( (3 (mod, (1.9 g 3 4 (mod 2, ( n 2 (4 + 3g 0 ( 2 n 1 C (1.11 for all n Z + {1, 2, 3,... }, where C denotes the Catalan number ( 2 /( + 1 ( ( (iii Provided > 5, we have g ( (mod, (1.12 g ( 1 ( 1 f ( 1 H 2 0 (mod 2, (1.13 ( 1 E 3 (mod. (1.14 Remar 1.1. Let > 3 be a rime. By [JV, Lemma 2.7], g 3 9 g 1 (mod for all 0,..., 1. So (1.9 imlies that 1 g ( 9 1 g We conecture further that 1 g 1 ( g 1 2 (3 1 (mod 2 and 0 g ( 9 3 (mod. (mod 2. In [S13b] the author showed the following congruences similar to (1.12 and (1.13: 1 ( 1 f 2 0 (mod and 1 ( 1 f 0 (mod 2.

5 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 5 Such congruences are interesting in view of Wolstenholme s congruences H 1 0 (mod 2 and H (2 1 0 (mod. Alying the Zeilberger algorithm (cf. [PWZ, ] via Mathematica 9 we find the recurrence for s n g n ( 1 (n 0, 1, 2,... : (n (4n + 5s n+3 + (20n n n + 165s n+2 + (76n n n + 375s n+1 25(n (4n + 9s n 0. In contrast with (1.11, we are also able to show the congruence (3 + 1 f ( (2 2 (mod 5 ( via the combinatorial identity 1 n 2 (3 + 1f 8 0 ( n ( 1 n n 2 ( (1.16 which can be shown by the Zeilberger algorithm. We are going to investigate in the next section connections among the olynomials A n (x, f n (x and g n (x. Section 3 is devoted to our roof of Theorem 1.1. In Section 4 we shall roose some conectures for further research. Obviously, 2. Relations among A n (x, f n (x and g n (x 1 (2 + 1 n Z and 1 n n 0 (2 + 1( 1 ( 1 Z for all n 1, 2, 3,.... This is a secial case of our following general result. Theorem 2.1. Let X n n 0 ( n ( n + 0 x and y n n 0 ( n x for all n N. (2.1 Then X n n 0 ( n ( n + ( 1 n y for every n N. (2.2

6 6 ZHI-WEI SUN Also, for any n Z + we have and ( 1 ( ( n 1 n + (2 + 1X ( 1 y (2.3 n ( 1 ( ( n 1 n + (2 + 1( 1 X x. (2.4 n Proof. If n N, then n ( ( n n + l ( 1 l y l l l l0 n ( ( l ( n n 1 l x l l l0 0 n ( n n ( ( n n 1 x n l l 0 l n ( ( n 1 x (by the Chu-Vandermonde identity [G, (2.1] n 0 n ( ( n n + ( 1 n x 0 0 and hence (2.2 holds. For any given integer 0, by induction on n we have ( ( l + n + ( 1 l (2l + 1 ( 1 (n 2 2 l (2.5 for all n + 1, + 2,.... Fix a ositive integer n. In view of (2.2 and (2.5, (2l + 1X l (2l + 1 l0 l0 l ( l ( 2 ( 1 l y ( 2 ( l + ( 1 y ( 1 l (2l l ( ( 2 n + ( 1 y ( 1 (n 2 0 ( ( n 1 n + ( 1 n ( 1 y. 0

7 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 7 This roves (2.3. Similarly, l ( l + (2l + 1( 1 l X l (2l + 1( 1 l 2 l0 l0 0 ( 2 x ( 2 ( l + x ( 1 l (2l l ( ( 2 n + x ( 1 (n 2 0 ( ( n 1 n + ( 1 n x. and hence (2.4 is also valid. Combining the above, we have comleted the roof of Theorem 2.1. Lemma 2.1. For any nonnegative integers m and n we have the combinatorial identity n ( ( ( ( ( m x + y n + x y x + x y. (2.6 n m + n m n 0 0 Remar 2.1. (2.6 is due to Nanundiah, see, e.g., (4.17 of [G,. 53]. The author [S12] roved that 1 n 0 (2 + 1A (x Z[x] for all n Z +, and conectured that 1 n 0 (2 + 1( 1 A (x Z[x] for any n Z +, which was confirmed by Guo and Zeng [GZ]. Theorem 2.2. Let n be any nonnegative integer. Then and A n (x n 0 ( n f (x g n (x, f n (x n 0 ( n ( n + Also, for any n Z + we have f (x n 0 ( n n 0 ( n ( 1 n g (x, (2.7 ( n + ( 1 n g (x. (2.8 ( 1 n ( ( n 1 n + (2 + 1A (x ( 1 g (x (

8 8 ZHI-WEI SUN and ( 1 n ( ( n 1 n + (2 + 1( 1 A (x f (x. ( Proof. By the binomial inversion formula (cf. (5.48 of [GKP,. 192], the two identities in (2.7 are euivalent. Observe that n l0 ( n f l (x l n ( l ( ( ( n l 2 x l l 0 n ( ( n 2 n ( ( n x n l l l n ( ( ( n 2 n x g n (x n l0 0 0 with the hel of the Chu-Vandermonde identity. Thus (2.7 holds. Next we show (2.8. Clearly n l0 ( n l ( n + l l f l (x n ( ( l ( ( ( n n + l l 2 x l l l 0 n ( ( n 2 n ( ( ( n n + l x l l n l0 0 n ( ( n 2 x 0 n 0 ( n ( 2 l ( ( n ( n + + n ( n + (by Lemma x ( n + n This roves the first identity in (2.8. Alying Theorem 2.1 with x n f n (x and X n A n (x for n N, we get the identity A n (x n 0 ( n ( n + as well as (2.9 and (2.10, with the hel of (2.7. The roof of Theorem 2.2 is now comlete. ( 1 n g (x (2.11 Remar 2.2. (2.7 and (2.8 in the case x 1 are well nown.

9 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 9 Corollary 2.1. Let be an odd rime. Then and 0 0 A (x 0 ( 1 f (x ( 1 g (x A (x (mod 2 (2.12 (mod 2. (2.13 Proof. In view of (2.8, A l (x l0 Similarly, l l ( + l 2 ( 2 f (x ( 2 f (x ( f (x ( 1 (mod 2. 0 ( 1 l A l (x l0 l ( 2 l ( + l 2 ( 2 ( 2 ( + l f (x 2 l f (x (2 + 1! ( 2 ( 1 g (x ( 1 g (x ( g (x (mod 2. 0 This concludes the roof of Corollary < ( 2 2 Remar 2.3. In [S12] the author investigated 1 0 (±1 A (x mod 2 (where is an odd rime and made some conectures. For any n Z we set [n] 1 n 1 { 0 <n if n 0, n 0 < n if n < 0; this is the usual -analogue of the integer n. Define [ ] n 1 and 0 [ ] n 1 [n + 1] [] for Z +.

10 10 ZHI-WEI SUN [ Obviously, lim n 1 ] ( n. For n N we define and Clearly A n (x; : g n (x; : n [ n 2n(n 0 ] 2 n [ n 2n(n 0 lim A n(x; A n (x and 1 ] 2 [ n + ] 2 x [ ] 2 x. lim g n (x; g n (x. 1 Those identities in Theorem 2.2 have their -analogues. For examle, the following theorem gives a -analogue of (2.11. Theorem 2.3. Let n N. Then we have A n (x; n [ ] [ ] n n + ( 1 n (n (5n+3+1/2 g (x;. ( Proof. Let {0,..., n}. By the -Chu-Vandermonde identity (see, e.g., Ex. 4(b of [AAR,. 542], n [ ] [ ] [ ] n 1 n 2 1 ( 2 n n This, together with yields that n [ ] n 1 [ ] n 1 ( 2 [ ] [ ] [ ] n 1 n 1 [ ] [ ] n,. [ ] [ ] n n. It is easy to see that [ ] m 1 [ ] m + ( 1 m (+1/2.

11 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 11 So we are led to the identity n ( 1 n (n +1 Since [ ] n [ ] 2 +2(n [ ] n [ ] n [ ] n + multilying both sides of (2.15 by [ n n ( 1 n (n (n [ ] n [ ] [ ] n [ ] [ ] n + n + 2 [ ] [ ] [ ] [ ] n n + n + 2 and, 2 ] [ 2 ] x we get [ ] [ ] 2 [ ] [ ] 2 [ n + 2 n n + x ] 2. (2.15 x. In view of the last identity we can easily deduce the desired (2.14. By alying Theorem 2.2 we obtain the following new result. Theorem 2.4. Let n be any ositive integer. Then ( 1 ( A 0 (mod n 3. ( Proof. By induction on n, for each 0,..., n 1 we have ( ( l + n + ( 1 l (6l 3 + 9l 2 + 5l + 1 ( 1 (n (3n l Thus, in view of (2.8, 1 ( 1 n l (6l 3 + 9l 2 + 5l + 1A l (x n l0 ( 1n l ( ( l + 2 ( 1 l (6l 3 + 9l 2 + 5l + 1 f (x n 2 l0 0 ( 1n ( 2 ( l + f (x ( 1 l (6l 3 + 9l 2 + 5l + 1 n 2l 0 l ( 1n ( ( 2 n + f (x( 1 (n (3n n 2 0 ( ( n 1 n + ( n 2 f (x. 0

12 12 ZHI-WEI SUN Hence we have reduced (2.16 to the congruence ( ( n 1 n + (3 + 2f 0 (mod n 2. ( The author [S13a, (1.12] conectured that a m : 1 m 1 m 2 (3 + 2( 1 f Z for all m 1, 2, 3,..., 0 and this was confirmed by V.J.W. Guo [Gu]. Set a 0 0. Observe that ( ( n 1 n + (3 + 2f 0 ( ( n 1 n 1 (( a +1 2 a 0 n ( ( n 1 n 1 ( ( n 1 n 1 2 a 2 a ( n 1 n 2 a n + (( ( ( ( n 1 n 1 n 1 n 1 2 a. n <<n As ( ( n 1 n ( ( n 1 n 1 n2 2 ( n 1 1 ( n 1 1 for all 1,...,, we have (2.17 by the above, and hence (2.16 holds. The author [S12] conectured that for any rime > 3 we have (2 + 1( 1 A 3 0 (mod 3, (2.18 and this was confirmed by Guo and Zeng [GZ]. Corollary 2.2. Let > 3 be a rime. Then ( ( 1 A (mod 3. (2.19

13 Proof. Clearly CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 13 3( ( ( Thus (2.19 follows from (2.16 and (2.18. Remar 2.4. Let > 3 be a rime. We are also able to rove that and ( ( 1 A ( ( 1 A (mod 3 (2.20 (mod 3. (2.21 It seems that for each r 0, 1, 2,... there is a -adic integer c r only deending on r such that ( r+1 ( 1 A c r (mod Proof of Theorem 1.1 Lemma 3.1. For any odd rime, we have 1 (2 + 1A (x 0 g (x 2 0 Proof. Obviously, ( ( 1 + ( 1 0< 0 (1 2 2 g (xh (2 (mod 4. ( H (2 (mod 4 (3.2 for every 0,..., 1. Thus (3.1 follows from (2.9 with n. Lemma 3.2. Let > 3 be a rime. Then g 1 (1 + 2 (3 3 (mod 2. (3.3 Proof. For 0,..., 1, clearly ( 2 1 ( 1 2 0< 0< ( 1 2 ( 2 1 ( 1 Thus, with the hel of [S12b, Corollary 2.2] we obtain ( ( (2 g 1 ( (mod and hence (3.3 holds. (mod 2.

14 14 ZHI-WEI SUN Lemma 3.3. For any odd rime, we have 0 ( 1/2 ( (mod 2. (3.4 Proof. Clearly (3.4 holds for 3. Below we assume > 3. Observe that ( 3 ( 1/ ( ( 3 ( 1/2 2(( 1/ ( 3 ( 1/2+ 2(( 1/ Since 1 we have 1 ( 3 1 ( ( ( 1/2 1 ( ( 3 ( ( 1/2 4 ( ( 1/2 1 ( 1 ( ( (2 (3 (mod. 1 3 ( ( ( 3 1 (mod for 1,..., 1, 4( (mod Note also that ( 1/2 1 ( 3 1 ( 1/ ( 3x 1 dx ( 3x ( 1/2 dx 1 + 3x 1 ( 1/2 ( ( 1/2 ( 1 3x 1 dx 0 1 ( ( 1/2 ( 1 3x x0 ( 1/2 ( 1 ( ( ( (2 (mod

15 since 1 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 15 ( (2 (mod and 1 ( 2 0 (mod 2 by [ST1, (1.12 and (1.20]. Thus, in view of the above, we get 0 ( 1/2 It follows that ( (2 + 1 ( (2 (3 (3 3 2 ( ( 3( 1/ We are done. ( 3 ( 1/2 ( 3 ( 1/2 (mod. ( 3 3 ( 1/2 + ( 3 2 ( ( 3 ( 1/2 Lemma 3.4. For any rime, we have ( 2 1 ( 3 3 ( ( 3 ( 1/2 (mod 2. ( 3 ( ( 1 (mod 2 for all 1,..., 1. (3.5 r r r0 Proof. Define ( ( 1 u r r r0 for all N. Alying the Zeilberger algorithm via Mathematica 9, we find the recurrence ( (2(2 + 1u +1 u ( ( 1 ( + ( + 1( ( ( ( Thus, for each 1,..., 2, we have 2(2 + 1u +1 u (mod 2

16 16 ZHI-WEI SUN and hence ( ( 2( ( + 1 u +1 2( ( 2 2(2 + 1 u +1 u +1 ( 2 u (mod 2. So it remains to rove ( 2 1 u1 (mod 2. With the hel of the Chu- Vandermonde identity, we actually have ( 2 u 1 ( 1 r r r0 ( 1 1 ( 3 1 This concludes the roof. ( Proof of Theorem 1.1. (i By [S12, (2.13], 1 0 (2 + 1A (x 0 ( r0 ( 1 1 r H (2 Combining this with (3.1 we immediately get (1.4. By [S12, (1.6-(1.7], and 1 (2 + 1A B 3 (mod (2 + 1A ( 1 0 ( 2 r x (mod 4. ( 1 2 E 3 (mod 3. Combining this with (3.1 we obtain (1.5 and (1.6. In view of (1.4 and (3.4, we get (1.7. (ii With the hel of (2.7, l1 g l (x l l l l 0 f (x ( l f (x H 1 + l ( l l l1 1 f (x ( 1 f (x(1 H (mod 2. f (x l ( 1 ( l

17 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 17 In view of [S13b, (2.7], this imlies that 1 g (x (+1/2 x ( 1 H f (x (mod 2. (3.6 So (1.8 follows. By induction, for any integers m > 0, we have m 1 ( n + (2n n m(m + 1 ( m + 2. This, together with (2.8 and (3.2, yields ( 1 n (2n + 1A n (2n + 1 n0 n0 0 0 ( 2 n ( n ( 2 ( 1 g ( n + ( 1 g (2n n ( 2 ( 1 ( g + 1 ( + ( g 1 ( ( 2 1 g g g g 1 (mod 4 2 ( ( 1 + ( 1 g + 1 since ( (mod 3 by Wolstenholme s theorem. Combining this with (2.18 and (3.3, we obtain (1 + 2 ( g 1 (mod 3 and hence (1.9 follows. (1.10 follows from a combination of (1.5 and (1.11 in the case n. If we let u n denote the left-hand side or the right-hand side of (1.11, then by

18 18 ZHI-WEI SUN alying the Zeilberger algorithm via Mathematica 9 we get the recurrence relation (n + 2(n (2n + 3u n+3 (n + 2(22n n n + 120u n+2 (n + 1(38n n n + 102u n+1 + 9n 2 (n + 1(2n + 5u n for n 1, 2, 3,.... Thus (1.11 can be roved by induction. (iii Now we show (1.12-(1.14 rovided > 5. Observe that l1 g l (x 1 l 2 l l l 2 1 ( 2 2 ( 2 2 x x ( l 2 ( x 1 ( 2 2 ( x 1 ( 2 (mod. ( l 1 1 l ( 2 2 x 2 1 Recall that H (2 1 0 (mod. Also, for any 1,..., 1 we have 1 0 ( 2 0 by the Chu-Vandermonde identity. Thus 0 ( ( 2( g (x 2 1 ( 2 2 x (( 2( 1 1 ( 2 2 x (mod (Note that ( 2 ( 2( 0 (mod for 1,..., 1. It is nown that 1 ( 1 (cf. Tauraso [T] and moreover ( 1/2 1 ( ( 2 0 (mod (3.7 ( B 3 (mod 2

19 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 19 by Sun [S14]. So (1.12 is valid. Note that l1 g l (x 1 l l1 1 1 l l 1 ( 2 x ( l 2 ( x 1 ( 2 x l ( ( ( l 1 1 For 1 1 and < 1, clearly ( ( ( + 1!( +! ( 1!!(! 2 0 (mod 2. If with 1 1, then ( ( ( ( (mod ( 1 Recall that H 1 0 (mod 2. So we have 1 ( g (x 2 ( ( ( x ( 2 ( ( 1 ( 1 2 x 2 x x 1 ( ( 2 2 x 2 x (mod with the hel of (3.5. Thus, in view of (3.7 we get 1 g ( 1 1 ( 1 2 ( 1/2 1 ( ( 1 This roves (1.13. Combining this with (3.6 we obtain 2 ( l + ( 1 ( 2 0 (mod 2. 1 ( 1 f ( 1 H (+1/2 2 ( 1 ( 1 2 ( 1/2 1 E 3 (mod ( 1 with the hel of [S11b, Lemma 2.4]. So (1.14 holds. In view of the above, we have comleted the roof of Theorem

20 20 ZHI-WEI SUN 4. Some oen conectural congruences In this section we ose several related conectural congruences. Conecture 4.1. (i For any integer n > 1, we have ( ( 1 f 0 (mod (n 1n 2 0 Also, for each odd rime we have ( ( 1 f 3 2 ( (2 (mod 4. 0 (ii For every n 1, 2, 3,..., we have and the number 1 (4 + 3g (x Z[x] n 0 1 n 2 ( g ( 1 0 is always an odd integer. Also, for any rime we have ( g ( (mod 3. 0 For any nonzero integer m, the 3-adic valuation ν 3 (m of m is the largest a N with 3 a m. For convenience, we also set ν 3 (0 +. Conecture 4.2. Let n be any ositive integer. Then ( ( ν 3 (2 + 1( 1 A 3ν 3 (n ν 3 ( ( 1 A. 0 If n is a ositive multile of 3, then 0 ( ν 3 ( ( 1 A 3ν 3 (n

21 CONGRUENCES INVOLVING g n (x n ( n 2 ( 2 0 x 21 Conecture 4.3. For n N define F n : n 0 For any n Z +, the number ( 3 n ( 8 and G n : n 0 1 (6 + 5( 1 F n 0 ( 2 n (6 + 1C. is always an odd integer. Also, for any rime > 3 we have ( 1 F 3 0 (mod 2 and G B 3 (mod 4. 1 Remar 4.1. For any rime > 3, the author [S13b, S12] roved that 1 0 ( 1 f 3 (mod 2 and 1 1 h 0 (mod 2 with h ( 2C 0. Acnowledgment. comments. The author would lie to than the referee for helful References [AAR] G. Andrews, R. Asey and R. Roy, Secial Functions, Cambridge Univ. Press, Cambridge, [B] P. Barrucand, A combinatorial identity, roblem 75-4, SIAM Review 17 (1975, 168. [C] D. Callan, A combinatorial interretation for an identity of Barrucand, J. Integer Se. 11 (2008, Article , 3. [G] H. W. Gould, Combinatorial Identities, Morgantown Printing and Binding Co., West Virginia, [GKP] R. L. Graham, D. E. Knuth and O. Patashni, Concrete Mathematics, 2nd ed., Addison-Wesley, New Yor, [Gu] V. J. W. Guo, Proof of two conectures of Sun on congruences for Franel numbers, Integral Transforms Sec. Funct. 24 (2013, [GZ] V. J. W. Guo and J. Zeng, Proof of some conectures of Z.-W. Sun on congruences for Aéry olynomials, J. Number Theory 132 (2012, [JV] F. Jarvis and H. A. Verrill, Suercongruences for the Catalan-Larcombe-French numbers, Ramanuan J. 22 (2010, [PWZ] M. Petovše, H. S. Wilf and D. Zeilberger, A B, A K Peters, Wellesley, Massachusetts, [Sl] N. J. A. Sloane, Seuence A in OEIS (On-Line Encycloedia of Integer Seuences, htt://oeis.org/a [St92] V. Strehl, Recurrences and Legendre transform, Sém. Lothar. Combin. 29 (1992, [St94] V. Strehl, Binomial identities combinatorial and algorithmic asects, Discrete Math. 136 (1994,

22 22 ZHI-WEI SUN [S11] Z.-W. Sun, List of conectural series for owers of π and other constants, rerint, arxiv: [S11a] Z.-W. Sun, On congruences related to central binomial coefficients, J. Number Theory 131 (2011, [S11b] Z.-W. Sun, Suer congruences and Euler numbers, Sci. China Math. 54 (2011, [S12] Z.-W. Sun, On sums of Aéry olynomials and related congruences, J. Number Theory 132 (2012, [S12a] Z.-W. Sun, Arithmetic theory of harmonic numbers, Proc. Amer. Math. Soc. 140 (2012, [S12b] Z.-W. Sun, On sums of binomial coefficients modulo 2, Collo. Math. 127 (2012, [S13a] Z.-W. Sun, Connections between x 2 +3y 2 and Franel numbers, J. Number Theory 133 (2013, [S13b] Z.-W. Sun, Congruences for Franel numbers, Adv. in Al. Math. 51 (2013, [S14] Z.-W. Sun, -adic congruences motivated by series, J. Number Theory 134 (2014, [ST1] Z.-W. Sun and R. Tauraso, New congruences for central binomial coefficients, Adv. in Al. Math. 45 (2010, [ST2] Z.-W. Sun and R. Tauraso, On some new congruences for binomial coefficients, Int. J. Number Theory 7 (2011, [T] R. Tauraso, More congruences for central binomial congruences, J. Number Theory 130 (2010, [vp] A. van der Poorten, A roof that Euler missed... Aéry s roof of the irrationality of ζ(3, Math. Intelligencer 1 (1978/79, [W] J. Wolstenholme, On certain roerties of rime numbers, Quart. J. Al. Math. 5 (1862, [Z] D. Zagier, Integral solutions of Aéry-lie recurrence euations,, in: Grous and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, [Zh] J. Zhao, Wolstenholme tye theorem for multile harmonic sums, Int. J. Number Theory 4 (2008,