Congruences involving binomial coefficients and Apéry-like numbers
Zhi-Hong Sun

Publ. Math. Debrecen 96(2020), no.3-4, 315-346.
Congruences involving binomial coefficients and  Apéry-like
numbers
Zhi-Hong Sun
 For n=0,1,2,… let W_n=∑_k=0^[n/3]2kk
 3kk n3k(-3)^n-3k,
 where [x] is the greatest integer not exceeding x. Then
 {W_n} is an Apéry-like  sequence. In this paper we deduce many congruences
 involving {W_n}, in particular we determine
∑_k=0^p-12kkW_k/m^k p for
m=-640332,-5292,-972,-108,-44,-27,-12,8,54,243 by using binary
quadratic forms, where p>3 is a prime. We also prove several
congruences for generalized Apéry-like numbers, and pose 29
challenging conjectures on congruences involving binomial
coefficients and Apéry-like numbers.
 MSC: Primary 11A07, Secondary
11A15,11B50,11B65,11B68,11E25,33C45,65Q30
 Keywords: Apéry-like numbers; binomial coefficients; congruence; binary
 quadratic form; Legendre polynomial; Bernoulli number
 The author is supported by
the National Natural Science Foundation of China (Grant No.
11771173).


§ 1. INTRODUCTION

For s>1 let ζ(s)=∑_n=1^∞ 1n^s.
In 1979, in order to prove ζ(2) and
 ζ(3) are irrational,
 Apéry [4] introduced the Apéry numbers {A_n} and {A'_n}
 given by
 
    A_n= ∑_k=0^n nk^2n+kk^2andA'_n=∑_k=0^nṉk^2ṉ+̱ḵk.

It is well known (see [7]) that

    (n+1)^3A_n+1=(2n+1)(17n(n+1)+5)A_n-n^3A_n-1 (n≥ 1),
    
       (n+1)^2A'_n+1=(11n(n+1)+3)A'_n+n^2A'_n-1 (n≥ 1).

Let Z and Z^+ be the set of integers and the set of
positive integers, respectively.  The first kind of Apéry-like
numbers {u_n} satisfies

    u_0=1,  u_1= b, (n+1)^3u_n+1=(2n+1)(an(n+1)+b)u_n-cn^3u_n-1 (n≥ 1), 1.1

where a,b,c∈ Z and c≠0. Let [x] be the greatest
integer not exceeding x, and let

    D_n=∑_k=0^nṉk^22̱ḵk2̱ṉ-̱2̱ḵn-k, T_n=∑_k=0^nṉk^22̱ḵn^2, 
        b_n=∑_k=0^[n/3]2̱ḵk3̱ḵkṉ3kṉ+̱ḵk(-3)^n-3k.

Then {A_n}, {D_n}, {b_n} and {T_n}  are the first
kind of Apéry-like numbers with
(a,b,c)=(17,5,1),(10,4,64),(-7,-3,81),(12,4,16), respectively. The
numbers {D_n} are called Domb numbers, and {b_n} are called
Almkvist-Zudilin numbers. For {A_n}, {D_n}, {b_n} and
{T_n} see A005259, A002895, A125143 and A290575 in Sloane's
database “The On-Line Encyclopedia of Integer Sequences" [40]. For
the congruences involving T_n see the author's recent paper [33].
In 2009, Zagier [41] studied the second kind of Apéry-like numbers
{u_n} given by

    u_0=1, u_1=band(n+1)^2u_n+1=(an(n+1)+b)u_n-cn^2u_n-1 (n≥ 1), 1.2

where a,b,c∈ Z and c≠0.  Let
 
    f_n=∑_k=0^nṉk^3=∑_k=0^nṉk^22̱ḵn,
       S_n=∑_k=0^[n/2]2̱ḵk^2ṉ2k4^n-2k=∑_k=0^nṉk2̱ḵk2̱ṉ-̱2̱ḵn-k,
     
       a_n=∑_k=0^nṉk^22̱ḵk,
      Q_n=∑_k=0^nṉk(-8)^n-kf_k,
     
       W_n=∑_k=0^[n/3]2̱ḵk3̱ḵkṉ3k(-3)^n-3k.

  In [41] Zagier stated that {A'_n}, {f_n}, {S_n}, {a_n},{Q_n}
and {W_n} are the second kind of Apéry-like sequences with
(a,b,c)=(11,3,-1),(7,2,-8),(12,4,32),(10,3,9),(-17,-6,72), (-9,-3,27), respectively. The  sequence
 {f_n} is called Franel numbers.
 In [14,31,32]  the author
 systematically investigated identities and congruences for sums
 involving S_n or f_n.
  For {A'_n}, {f_n}, {S_n}, {a_n},{Q_n}
and {W_n}, see A005258, A000172, A081085, A002893, A093388 and
A291898 in Sloane's database [40].
Apéry-like numbers have fascinating properties and they are
concerned with Picard-Fuchs differential equation, modular forms,
hypergeometric series, elliptic curves, series for 1π,
supercongruences, binary quadratic forms, combinatorial identities,
Bernoulli numbers and Euler numbers. See typical papers
[1,2,3,6,8,9,10,11,13,17,18,21,30,35,36,39].
For a∈ Z and given odd prime p let ap be the
Legendre symbol. For a prime p let Z_p be the set of
rational numbers whose denominator is not divisible by p. For
positive integers a,b and n, if n=ax^2+by^2 for some integers
x and y, we briefly write that n=ax^2+by^2.

In Section 2 we obtain some congruences for sums involving W_n.
 From [9, (6.4)] we know
that

    (∑_k=0^∞W_kx^k)^2= 11-27x^2∑_k=0^∞2̱ḵk(x(1+9x+27x^2)(1-27x^2)^2)^kW_k.
    1.3
 We prove the p-analogue of (1.3):

    (∑_k=0^p-1W_kx^k)^2
    ∑_k=0^p-12̱ḵkx(1+9x+27x^2)(1-27x^2)^2^kW_k
    p, 1.4
 where p>3 is a prime, x∈ Z_p and
(x+3)(1+9x+27x^2)(1+9x)(1+27x^2)(1-27x^2)0 p.
Suppose that p>3 is a prime, n∈ Z_p and n(n-12)0 p. In Section 2, we show that

    ∑_k=0^p-12̱ḵkW_k(n-12)^kn(n-12)p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3kn^3k p. 1.5
 As consequences, we determine
∑_k=0^p-12̱ḵkW_km^k p for
m=-640332,-5292,-972,-108, -44,-27,8,54,243 by using binary
quadratic forms. We also determine W_p-12,
∑_k=0^p-1W_k(-3)^k, ∑_k=0^p-1W_k(-9)^k
and ∑_k=0^p-12̱ḵkW_k(-12)^k modulo p.

Let p be an odd prime. In 2010 Jarvis and Verrill [13] discovered relations between u_n
and u_p-1-n modulo p for u_n=A_n',a_n,f_n or S_n. For
example, they proved f_n (-8)^nf_p-1-n p for
n=0,1,…,p-1. In Section 3 we establish a vast generalization
of such congruences
 for generalized Apéry-like numbers {u_n}. See
Theorem 3.1.
In [20] S. Ramanujan made some conjectures for 1/π, which
involve the following four sums

    ∑_k=0^∞(ak+b)2̱ḵk^3m^k,∑_k=0^∞(ak+b)2̱ḵk^23̱ḵkm^k,∑_k=0^∞(ak+b)2̱ḵk^24̱ḵ2km^k,∑_k=0^∞(ak+b)2̱ḵk3̱ḵk6̱ḵ3km^k.

 The author's brother Z.W. Sun in [39] and the author in [25] posed many
conjectures on congruences for

    ∑_k=0^p-12̱ḵk^3m^k,∑_k=0^p-12̱ḵk^23̱ḵkm^k,∑_k=0^p-12̱ḵk^24̱ḵ2km^k,∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3km^kp^2,
    1.6

 where m∈ Z and p is an odd prime with p∤ m.
Some of such conjectures were proved by the author in [27-29]. In
particular, most of conjectures were solved when the modulus is p.
Let p>3 be a prime. Recently Liu [16] conjectured congruences for

    ∑_k=0^p-12̱ḵk^23̱ḵk108^k,∑_k=0^p-12̱ḵk^24̱ḵ2k256^k,∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k1728^kp^3
 in
terms of p-adic gamma functions.  Based on calculations with
Maple, in Section 4 we pose 29 challenging conjectures on
congruences involving sums in (1.6) or Apéry-like numbers. See
Conjectures 4.1-4.29.



§ 2. CONGRUENCES FOR SUMS INVOLVING W_N

For any nonnegative integer n, define

    W_n(x)=∑_k=0^[n/3]2̱ḵk3̱ḵkṉ3kx^n-3k.

Then W_n=W_n(-3). In this section we establish some congruences
involving W_n and W_n(x) modulo a prime.
 We begin with three useful lemmas.
 Lemma 2.1 Let n be a nonnegative
integer. Then

    ∑_k=0^nṉkW_k(x)y^n-k=W_n(x+y).

Proof. It is clear that

    ∑_k=0^nṉkW_k(x)y^n-k   =∑_k=0^n
    ṉky^n-k∑_r=0^k2̱ṟr3̱ṟrḵ3rx^k-3r
       =∑_r=0^n2̱ṟr3̱ṟry^n-3r∑_k=r^nṉkḵ3r xy^k-3r
       =∑_r=0^n2̱ṟr3̱ṟry^n-3r∑_k=3r^nṉ3rṉ-̱3̱ṟk-3r xy^k-3r
       =∑_r=0^n2̱ṟr3̱ṟrṉ3ry^n-3r(1+ xy)^n-3r=W_n(x+y).
 This proves the lemma.
Let {P_n(x)} be the famous Legendre polynomials given by

    P_n(x)   =∑_k=0^nṉkṉ+̱ḵkx-12^k
    = 12^n∑_k=0^[n/2]ṉk(-1)^k2̱ṉ-̱2̱ḵnx^n-2k.

 It is well known that
 
    P_0(x)=1, P_1(x)=x, (n+1)P_n+1(x)=(2n+1)xP_n(x)-nP_n-1(x)(n≥ 1).

Lemma 2.2 Suppose that p>3 is a prime and m,x∈ Z_p
with mx0 p. Then

    ∑_k=0^p-1W_k(x+m)m^k    W_p-1(x)
    ∑_k=0^p-12̱ḵk3̱ḵk(-x)^3k P_[ p3](1+54x^3)
    
        - p3∑_n=0^p-1n^3-3x(x^3-216)n
    -2x^6-1080x^3+108^2p  p.

 Proof. Since p̱-̱1̱k (-1)^k p, using Lemma 2.1 and Fermat's
little theorem we see that

    ∑_k=0^p-1W_k(x+m)m^k∑_k=0^p-1p̱-̱1̱kW_k(x+m)(-m)^p-1-k=W_p-1(x) p.
 On the other hand,
since p|2̱ḵk3̱ḵk for p3<k<p we have

    W_p-1(x)=∑_k=0^[p-13]2̱ḵk3̱ḵkp̱-̱1̱3kx^p-1-3k∑_k=0^p-12̱ḵk3̱ḵk(-x)^3k p.
 By [28, Corollary 3.1],

    ∑_k=0^p-12̱ḵk3̱ḵk(-x)^3k    P_[ p3](1+54x^3)  -
    p3∑_n=0^p-1n^3-3(1-216x^3)n+
    108^2x^6-1080x^3-2p
    
       =- p3∑_n=0^p-1 nx^2^3
    -3(1-216x^3) nx^2+ 108^2x^6-1080x^3-2p
    
       =- p3∑_n=0^p-1n^3-3x(x^3-216)n
    -2x^6-1080x^3+108^2p p.

Thus the lemma is proved.

Let p>3 be a prime. Taking m=1 and x=-4 in Lemma 2.2 yields

    ∑_k=0^p-1W_k
    --6p∑_n=0^p-1n^3-840n+9074p p. 2.1
 Taking
m=-1 and x=-2 in Lemma 2.2 yields

    ∑_k=0^p-1(-1)^kW_k
    --6p∑_n=0^p-1n^3-336n+2522p p. 2.2



Lemma 2.3 For any nonnegative integer n we have

    ∑_k=0^nṉkW_k3^n-k=2̱ṉ/̱3̱n/3ṉn/3   if 3| n,
    0   if 3∤ n.

Proof. Putting x=-3 and y=3 in Lemma 2.1 gives

    ∑_k=0^nṉkW_k3^n-k=W_n(0)=∑_k=0^[n/3]2̱ḵk3̱ḵkṉ3k0^n-3k.
 This yields the result.
Now we are ready to prove the following result.
Theorem 2.1 Let p be a prime with p>3. Then

    ∑_k=0^p-1W_k(-3)^k∑_k=0^p-1W_k(-9)^k
        -L p   if p 1 3 and so 4p=L^2+27M^2 with L
    1 3,
    0 p   if p 2 3

and

    ∑_k=0^p-12̱ḵkW_k(-12)^k L^2 p   if p 1 3 and so 4p=L^2+27M^2,
    0 p   if p 2 3.

Proof. Putting m=-9 and x=6 in Lemma 2.2 we get
∑_k=0^p-1W_k(-9)^k P_[ p3]( 54) p. Now
applying [28, Theorem 3.2] gives the congruence for
∑_k=0^p-1W_k(-9)^k p. Since p̱-̱1̱k
(-1)^k p for k=0,1,…,p-1, using Lemma 2.3 and [5,
Theorem 9.2.1] we see that

    ∑_k=0^p-1W_k(-3)^k   ∑_k=0^p-1p̱-̱1̱kW_k3^p-1-k
       =2̱(̱p̱-̱1̱)̱3̱p-13p̱-̱1̱p-132̱(̱p̱-̱1̱)̱3̱p-13 -L p
     if 3| p-1
    and 4p=L^2+27M^2 with L 1 3,
    0 pif 3| p-2.

Note that p|2̱ḵk for k=p+12,…,p-1
and p̱-̱1̱2̱k-̱ ̱1̱2̱k=(-4)^-k2̱ḵk p for
k=0,1,…,p-12. Using Lemma 2.3 we see that

    ∑_k=0^p-12̱ḵkW_k(-12)^k   ∑_k=0^(p-1)/2p̱-̱1̱2̱kW_k3^k3p
    ∑_k=0^(p-1)/2p̱-̱1̱2̱k W_k· 3^p-12-k
       = 3pp̱-̱1̱3̱p-16p̱-̱1̱2̱p-16 2^p-13p̱-̱1̱2̱p-16^2 p   if 3| p-1,
    0 p   if 3| p-2.

Now assume p 1 3. Then p=A^2+3B^2 and 4p=L^2+27M^2 with
A,B,L,M∈ Z and A L 1 3. By [5, p.201],
p̱-̱1̱2̱p-16 2A p. If 2 is a cubic residue of
p, then 2^p-13 1 p. It is  well known that 3| B
and so L=-2A. Hence

    ∑_k=0^p-12̱ḵkW_k(-12)^k 2^p-13p̱-̱1̱2̱p-16^2 (2A)^2 L^2 p.
  Now assume that
2 is  a cubic nonresidue of p. Then 2^p-131
p, 3∤ B and 2∤ LM. We choose the sign of M so that
M L 4 and  the sign of B so that B A 1 3. From
[23, p.227] we know that

    2^p-13-1-A/B2 p,
    A=L-9M4andB=L+3M4.
 Hence

    ∑_k=0^p-12̱ḵkW_k(-12)^k    2^p-13p̱-̱1̱2̱p-16^2-1- AB2· 4A^2
    -1- AB2· 4(-3B^2)
    
       =6(A+B)B=6(L-9M4+L+3M4)L+3M4
    
       = 14(3L^2-27M^2) 14(3L^2+L^2)=L^2 p.

This proves the remaining part and the proof is now complete.
Remark 2.1 In [37, Conjecture 1.4] Zhi-Wei Sun
conjectured that if p is a prime such that p 1 6 and so
4p=L^2+27M^2 with L 1 3, then

    ∑_k=0^p-1W_k(-9)^k∑_k=0^p-1W_k(-3)^k -L+  pLp^2;
 if p is a prime with p 5 6, then
∑_k=0^p-1W_k(-9)^k 0p^2.
 Lemma 2.4
Let p be an odd prime, n,x∈ Z_p and n(n+4x)0
p. Then

    n+4xp∑_k=0^p-12̱ḵkW_k(x)(n+4x)^k-1p
    W_p-12(- n4)np∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3kn^3k p.

Proof. As p̱-̱1̱2̱k-̱ ̱1̱2̱k=2̱ḵk(-4)^-k p and
p|2̱ḵk 3̱ḵk6̱ḵ3k for p6<k<p, using Lemma 2.1
we see that

    ∑_k=0^p-12̱ḵkW_k(x)(n+4x)^k
       ∑_k=0^(p-1)/2p̱-̱1̱2̱kW_k(x)-4n+4x^k
    -4(n+4x)p∑_k=0^(p-1)/2p̱-̱1̱2̱kW_k(x)
    n+4x-4^p-12-k
       =-n-4xpW_p-12(- n4)
    =-n-4xp∑_k=0^[p/6]2̱ḵk3̱ḵkp̱-̱1̱2̱3k(- n4)^p-12-3k
       n(n+4x)p∑_k=0^[p/6]2̱ḵk3̱ḵk6̱ḵ3k
    1(-4)^3k· (-n/4)^3k
       n(n+4x)p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3kn^3k p.

This proves the theorem.
Theorem 2.2 Let p be an odd prime. Then

    W_p-12 4x^2 p   if p 1 4 and so
    p=x^2+y^2 with 2∤ x,
    0 p   if p 3 4.

Proof. Since W_1=-3 we see that the result is true for p=3. Now
assume p>3. Putting n=12 in Lemma 2.4 yields

    W_p-12=W_p-12(-3)-12p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k12^3k p.
 In [19] Mortenson
proved the congruence

    ∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k12^3k-3p4x^2
    p   if 4| p-1 and so p=x^2+y^2 with 2∤ x,
    0 p   if 4| p-3,

which was conjectured by Rodriguez-Villegas in 2003. Now combining
the above gives the result.
Theorem 2.3 Let p be a prime with
p>3. Then

    ∑_k=0^p-12̱ḵkW_k54^k p3(4x^2-2p) p   if p=x^2+4y^2 1 4,
    0 p   if p 3 4.
Proof. It is easy to check the result for p=11. Now assume that
p≠11. Taking n=66 and x=-3 in Lemma 2.4 and then applying
[29, Theorem 4.3] deduces the result.
Theorem 2.4 Let p>3 be a prime. Then

    ∑_k=0^p-12̱ḵkW_k8^k 4x^2-2p p   if p=x^2+2y^2 1,3 8,
    0 p   if p 5,7 8.
Proof. It is easy to check the result for p=5. Now assume that
p>5. Taking n=20 and x=-3 in Lemma 2.4 and then applying [29,
Theorem 4.4] we deduce the result.
Theorem 2.5 Let p be a prime with p≠2,3,7. Then

    -3p∑_k=0^p-12̱ḵkW_k(-27)^k-3p∑_k=0^p-12̱ḵkW_k243^k
        4x^2-2p
    p   if p=x^2+7y^2 1,2,4 7,
    0 p   if p 3,5,6 7.


Proof. It is easy to check the result for p=5,17. Now assume that
p≠5,17. Taking n=-15 and x=-3 in Lemma 2.4 and then
applying [29, Theorem 4.7] we deduce the congruence for
∑_k=0^p-12̱ḵkW_k(-27)^k p. Taking n=255 and
x=-3 in Lemma 2.4 and then applying [29, Theorem 4.7] we deduce
the remaining part.
Theorem 2.6 Let p be a prime and p≠2,11. Then

    ∑_k=0^p-12̱ḵkW_k(-44)^k x^2-2p p   if p11=1 and so 4p=x^2+11y^2,
    0 p   if p11=-1.
Proof. It is easy to check the result for p=3. Now assume that
p≠3. Taking n=-32 and x=-3 in Lemma 2.4 and then applying
[29, Theorem 4.8] we deduce the result.
 Theorem
2.7 Let p be a prime with p≠2,3,19. Then

    -3p∑_k=0^p-12̱ḵkW_k(-108)^k x^2-2p p   if p19=1 and so
     4p=x^2+19y^2,
    0 p   if p19=-1.
Proof. Taking n=-96 and x=-3 in Lemma 2.4 and then applying [29,
Theorem 4.9] we deduce the result.
Using Lemma 2.4 and [29, Theorem 4.9] one can also deduce the
following results.

Theorem 2.8 Let p be a prime, p≠2,3,43. Then

    -3p∑_k=0^p-12̱ḵkW_k(-972)^k x^2-2p p   if p43=1 and so
     4p=x^2+43y^2,
    0 p   if p43=-1.


Theorem 2.9 Let p be a prime with p≠2,3,7,67. Then

    -3p∑_k=0^p-12̱ḵkW_k(-5292)^k x^2-2p p   if p67=1 and so
     4p=x^2+67y^2,
    0 p   if p67=-1.


Theorem 2.10 Let p be a prime with  p≠2,3,7,11,163.
Then

    -3p∑_k=0^p-12̱ḵk
    W_k(-640332)^k
        x^2-2p p   if p163=1 and so
     4p=x^2+163y^2,
    0 p   if p163=-1.



Theorem 2.11 Suppose that p>3 is a prime, x∈ Z_p and
x(x^3+27)(x^3-216)(x^2+6x-18)0 p. Then

    W_p-1(x)^2   (∑_k=0^p-1W_k(-x-3)^k)^2
    ∑_k=0^p-12̱ḵk^23̱ḵk(-x^3+27x^6)^k
    
       x(x^3-216)p
    ∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-x^3+27x(x^3-216))^3k
       x^3+27pW_p-12x(x^3-216)4(x^3+27)
       ∑_k=0^p-12̱ḵk(-x^3+27(x^2+6x-18)^2)^kW_k p.

Proof. From Lemma 2.2 we see that
W_p-1(x)∑_k=0^p-1W_k(-x-3)^k∑_k=0^p-12̱ḵk3̱ḵk(-x)^3k p. By [28,
Theorem 2.1],

    (∑_k=0^p-12̱ḵk3̱ḵkm^k)^2∑_k=0^p-12̱ḵk^23̱ḵk(m(1-27m))^kp^2.
 Hence,

    W_p-1(x)^2
       (∑_k=0^p-1W_k(-x-3)^k)^2(∑_k=0^p-12̱ḵk3̱ḵk(-x)^3k)^2
    ∑_k=0^p-12̱ḵk^23̱ḵk(-
    1x^3(1+27x^3))^k p.
 By [30,
Theorem 2.2], for t∈ Z_p with 4t± 5 p,

    ∑_k=0^p-12̱ḵk^23̱ḵk1-t^2108^k
    4t+5p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(t+1)(1-t)^3432(4t+5)^3^k
    p.
 Taking t=-1-54x^3 gives

    ∑_k=0^p-12̱ḵk^23̱ḵk(-
    1x^3(1+27x^3))^k
       x(x^3-216)p
    ∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-x^3+27x(x^3-216))^3k p.
 Set
n=-x(x^3-216)x^3+27. Then
n-12=-x^4+12x^3-216x+324x^3+27=-(x^2+6x-18)^2x^3+27.
From Lemma 2.4 (with x=-3) and the fact (̱p̱-̱1̱)̱/̱2̱r
4^-r2̱ṟr p for 0≤ r≤p-12 we see that

    ∑_k=0^p-12̱ḵk(-x^3+27(x^2+6x-18)^2)^kW_k
    
       n-12p(-1)^p-12W_p-12(- n4)
    =x^3+27pW_p-12x(x^3-216)4(x^3+27)
       =x^3+27p∑_k=0^p-12̱ḵk3̱ḵkp̱-̱1̱2̱3kx(x^3-216)4(x^3+27)^p-12-3k
       x(x^3-216)p
    ∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-x^3+27x(x^3-216))^3k p.
 Now putting all the above
together proves the theorem.
Corollary 2.1  Suppose that p>3 is a prime, x∈ Z_p
and (x+3)(1+9x+27x^2)(1+9x)(1+27x^2)(1-27x^2)0 p. Then

    (∑_k=0^p-1W_kx^k)^2
    ∑_k=0^p-12̱ḵkx(1+9x+27x^2)(1-27x^2)^2^kW_k
    p.

Proof. Substituting x with - 1x-3 in Theorem 2.11 yields

    (∑_k=0^p-1W_kx^k)^2
       ∑_k=0^p-12̱ḵk(--( 1x+3)^3+27(
    1x^2-27)^2)^kW_k
    
       =∑_k=0^p-12̱ḵkx(1+9x+27x^2)(1-27x^2)^2^kW_k
    p.
 This proves the corollary.
We remark that Corollary 2.1 is the p-analogue of (1.3).



§ 3. CONGRUENCES FOR GENERALIZED APÉRY-LIKE NUMBERS

In 2010, Jarvis and Verrill [13] established a relation between u_n
and u_p-1-n modulo p for {A_n'},{a_n},{f_n} and
{S_n}, where p is an odd prime. Inspired by (1.1) and (1.2),
we introduce generalized Apéry-like numbers and prove a vast
generalization of those congruences given in [13].

 Theorem 3.1 Suppose r∈ Z^+ and c∈ Z
with c≠0. Let b(n) be the polynomial of n with integral
coefficients and the property b(-1-n)=(-1)^rb(n) for any n∈
Z. Define the sequence {u_n} by

    u_0=1, u_1=b(0)and (n+1)^ru_n+1=b(n)u_n-cn^ru_n-1(n≥ 1). 3.1
 Suppose that p is an odd prime with p∤ c
and u_p∈ Z_p. For n=0,1,2,…,p-1 we have

    u_n u_p-1c^nu_p-1-n cpc^nu_p-1-n p   if p∤ u_p-12,
     (-1)^r-1 cpc^nu_p-1-n p   if p| u_p-12.

In particular,

    u_p-1 cp p   if p∤ u_p-12,
     (-1)^r-1 cp p   if p| u_p-12.

Proof. By (3.1), for n∈{0,1,…,p-1}, u_n∈ Z_p,
(p-n)^ru_p-n=b(p-1-n)u_p-1-n-c(p-1-n)^ru_p-2-n and so
(-n)^ru_p-n b(-1-n)u_p-1-n-c(-n-1)^ru_p-2-n p. Since
b(-1-n)=(-1)^rb(n) we get n^ru_p-n
b(n)u_p-1-n-c(n+1)^ru_p-2-n p.
 Multiplying c^n on both sides gives

    (n+1)^rc^n+1u_p-2-n b(n)c^nu_p-1-n-cn^r·
    c^n-1u_p-n p. 3.2
 By (3.1),
p^ru_p=b(p-1)u_p-1-c(p-1)^ru_p-2. Thus b(p-1)u_p-1
c(-1)^ru_p-2 p. Since b(p-1) b(-1)=(-1)^rb(0) p we
see that b(0)u_p-1 cu_p-2 p.
 If p| u_p-1, we must have p| u_p-2 and
so p| u_p-3 by (3.1). If u_p-(m-1) u_p-m 0 p
for m∈{2,3,…,p-1}, then u_p-(m+1) 0 p by (3.1).
Hence  u_p-2 u_p-3⋯ u_1 u_0 0 p. But
u_0=1. This is a contradiction. Therefore p∤ u_p-1. Set
v_n=c^nu_p-1-n/u_p-1. Then v_0=1=u_0 and
v_1=cu_p-2/u_p-1 b(0)=u_1 p. By (3.2), for
n=1,2,…,p-1 we have (n+1)^rv_n+1 b(n)v_n-cn^rv_n-1
p. Hence u_n v_n=c^nu_p-1-n/u_p-1 p for
n=0,1,…,p-1 and so u_p-1 c^p-1u_0/u_p-1 p,
which implies u_p-1^2 c^p-1 1 p and so u_p-1ε_p p for ε_p∈{1,-1}. This yields
u_n c^nu_p-1-n/u_p-1ε_pc^nu_p-1-n p.
Taking n=p-12 gives u_p-12ε_pc^p-12u_p-12ε_p
cpu_p-12 p. Hence, if p∤ u_p-12, then
ε_p cp 1 p, ε_p= cp and so
u_n cpc^nu_p-1-n p. Now assume that p|
u_p-12. By the above argument, u_p+12ε_p
c^p+12u_p-32ε_p c cpu_p-32
p. By (3.1),

    p+12^ru_p+12=bp-12u_p-12-cp-12^ru_p-32 -cp-12^ru_p-32 p.
 Namely, u_p+12
(-1)^r-1cu_p-32 p. Hence c(ε_p
cp-(-1)^r-1)u_p-32 u_p+12-u_p+12=0 p. If
p| u_p-32, since p| u_p-12 we see that
u_p-52⋯ u_0 0 p by (3.1). But u_0=1.
Therefore p∤ u_p-32 and so ε_p
cp=(-1)^r-1. This yields u_nε_pc^nu_p-1-n=(-1)^r-1 cpc^nu_p-1-n p,
which completes the proof.
Corollary 3.1 Let p>3 be a
prime and n∈{0,1,…,p-1}. Then

    P_n(x) P_p-1-n(x) p, A_n A_p-1-n p,
    
       D_n 64^nD_p-1-n p, b_n 81^nb_p-1-n p,
       T_n 16^nT_p-1-n p,
     W_n p3 27^nW_p-1-n p,
    
       Q_n p3 72^nQ_p-1-n p.

Proof. Taking u_n=P_n(x),A_n,D_n,b_n,T_n in Theorem 3.1 yields the
first five congruences. Since p̱-̱1̱m (-1)^m p, using [25,
Corollary 2.2] we deduce that

    W_p-1=∑_k=0^[p/3]2̱ḵk3̱ḵkp̱-̱1̱3k(-3)^p-1-3k∑_k=0^[p/3]2̱ḵk3̱ḵk27^k p3 p.
 Recall
that (n+1)^2W_n+1=(-9n(n+1)-3)W_n-27n^2W_n-1 (n≥ 1).
Applying Theorem 3.1 yields the result for W_n. Using [32, Lemma
2.4] and [25, Corollary 2.2] we see that

    Q_p-1=∑_k=0^p-1p̱-̱1̱k(-8)^p-1-kf_k
    ∑_k=0^p-1f_k8^k∑_k=0^p-12̱ḵk3̱ḵk27^k p3 p.
 Recall
that (n+1)^2Q_n+1=(-17n(n+1)-6)Q_n-72n^2Q_n-1 (n≥ 1).
Applying Theorem 3.1 yields the result for Q_n.
Theorem 3.2
Let {u_n} be given in Theorem 3.1, and let p be an odd prime.
Suppose that u_m∈ Z_p for m=0,1,2,… and k∈
Z^+. Then

    u_kp+n u_kpu_n pforn=0,1,…,p-1.

Proof. We prove the theorem by induction on n. Clearly the result
is true for n=0 since u_0=1. By (3.1),
(kp+1)^ru_kp+1=b(kp)u_kp-c(kp)^ru_kp-1. Thus, u_kp+1
b(kp)u_kp b(0)u_kp=u_1u_kp p. This shows that the
result is also true for n=1. Now assume 2≤ m≤ p-1 and the
result holds for n<m. From (3.1) and the inductive hypothesis we
see that

    m^ru_kp+m    b(kp+m-1)u_kp+m-1-c(kp+m-1)^ru_kp+m-2
       
    b(m-1)u_kp+m-1-c(m-1)^ru_kp+m-2
        b(m-1)u_kpu_m-1-c(m-1)^ru_kpu_m-2
    =u_kp· m^ru_m p.
 Since p∤ m we get
u_kp+m u_kpu_m p. This shows that the result is true for
n=m. Hence the theorem is proved by induction.
Corollary
3.2 Let {u_n} be given in Theorem 3.1, and let p be an odd
prime. Suppose u_m∈ Z_p and u_mp u_m p for
m=1,2,3,…. For n∈ Z^+ write
n=n_0+n_1p+⋯+n_sp^s, where
n_0,n_1,…,n_s∈{0,1,…,p-1}. Then we have the Lucas
congruence u_n u_n_0u_n_1⋯ u_n_s p.
Proof. Set k=n_1+n_2p+⋯+n_sp^s-1. Then n=kp+n_0. By
Theorem 3.2,

    u_n    u_kpu_n_0 u_ku_n_0
    =u_n_1+n_2p+⋯+n_sp^s-1u_n_0
    u_n_0u_n_1u_(n_2+n_3p+⋯+n_sp^s-2)p
        u_n_0u_n_1u_n_2+n_3p+⋯+n_sp^s-2 u_n_0u_n_1u_n_2u_n_3+n_4p+⋯ +n_sp^s-3
       ⋯ u_n_0u_n_1⋯ u_n_s-2u_n_s-1+n_sp
    u_n_0u_n_1⋯ u_n_s-1u_n_s p.

 This proves the corollary.
 Remark 3.1 Suppose that p is a prime. For m,n∈ Z^+ with m≤ n
write n=n_0+n_1p+⋯+n_sp^s and m=m_0+m_1p+⋯+m_sp^s,
where n_0,…,n_s,m_0,…,m_s∈{0,1,…,p-1}. Then ṉmṉ_̱0̱m_0ṉ_̱1̱m_1⋯ṉ_̱s̱m_s p. This is
called Lucas theorem. From [13] and [17] we know that many
Apéry-like numbers satisfy the Lucas congruences.
Theorem
3.3 Let {u_n} be given by (3.1). Then

    ∑_k=0^n-1b(k)(-c)^n-1-ku_k^2=n^ru_nu_n-1(n=1,2,3,…).

Thus, if p is an odd prime, p∤ c and u_m∈ Z_p for
m∈ Z^+, then 
    ∑_k=0^p-1b(k)(-c)^ku_k^2
    0p^r.

Proof. Since

    (k+1)^ru_k+1(-c)^k-k^ru_k-1(-c)^k-1
    =(k+1)^ru_k+1+ck^ru_k-1(-c)^k=b(k)u_k(-c)^k,
 we
see that

    ∑_k=0^n-1b(k)(-c)^ku_k^2
    =∑_k=0^n-1((k+1)^ru_k+1u_k(-c)^k-k^ru_ku_k-1(-c)^k-1)
    =n^ru_nu_n-1(-c)^n-1.
 This yields the result.
As an example, taking u_n=P_n(x) in Theorem 3.3 gives

    ∑_k=0^n-1(-1)^n-1-k(2k+1)P_k(x)^2
    =nP_n(x)P_n-1(x)x. 3.3



§ 4. CONJECTURES ON CONGRUENCES INVOLVING BINOMIAL
COEFFICIENTS AND APÉRY-LIKE NUMBERS

The Bernoulli numbers
{B_n}, Euler numbers {E_n} and the sequence {U_n} are
defined by

    B_0=1,∑_k=0^n-1ṉkB_k=0(n≥ 2),
    
        E_0=1, E_n=-∑_k=1^[n/2]ṉ2kE_n-2k(n≥ 1),
    
        U_0=1, U_n=-2∑_k=1^[n/2]ṉ2kU_n-2k(n≥ 1).

For congruences involving B_n,E_n and U_n see [22,24,26].
Based on calculations with Maple, we pose the following
challenging conjectures:

Conjecture 4.1 Let p>3 be a prime. Then

    A_p-1 1+ 23p^3B_p-3p^4,
     D_p-1 64^p-1-p^36B_p-3p^4,
    
       b_p-1 81^p-1- 227p^3B_p-3p^4,
     T_p-1 16^p-1+p^34B_p-3p^4.

Remark 4.1 In [33] the author proved that
T_p-1 16^p-1p^3 for any prime p>3.
Conjecture
4.2 Let p>3 be a prime. Then

    A'_p-1 1+ 53p^3B_p-3p^4,
      f_p-1 8^p-1+ 58p^3B_p-3p^4,
    
        S_p-1
    (-1)^p-1232^p-1+p^2E_p-3p^3,   a_p-1
    p39^p-1+p^2U_p-3p^3,
    
        W_p-1 p327^p-1+p^2U_p-3p^3,
      Q_p-1 p372^p-1+ 52p^2U_p-3p^3.

Remark 4.2 Let p>3 be a prime. In [38] Z.W. Sun proved a congruence equivalent
to f_p-1 8^p-1p^3. In [33] the author proved a
congruence equivalent to S_p-1 (-1)^p-1232^p-1p^2.
Let p be an odd prime. In 2000 Ahlgren and Ono[2] proved
Beukers' conjecture A_p-12 c(p)p^2, where
{c(n)} is given by

    Φ_1(q)=q∏_k=1^∞(1-q^2k)^4(1-q^4k)^4=∑_n=1^∞
    c(n)q^n(|q|<1).
 It is well known that Φ_1( e^2π
iz) is a newform in S_4(Γ_0(8)). For |q|<1 define

    Φ_4(q)=q∏_k=1^∞(1-q^4k)^6=∑_n=1^∞α(n)q^n,
    
       Φ_2(q)=q∏_k=1^∞(1-q^k)^2(1-q^2k)(1-q^4k)
    (1-q^8k)^2 =∑_n=1^∞β(n)q^n,
    
       Φ_3(q)=q∏_k=1^∞(1-q^2k)^3(1-q^6k)^3=
    ∑_n=1^∞γ(n)q^n.

It is known that Φ_4( e^2π iz) is a weight 3
newform with complex multiplication by Q(√(-1)), and for
m∈{2,3} Φ_m( e^2π iz) is a weight 3 newform
with complex multiplication by Q(√(-m)). More precisely,

    Φ_4( e^2π iz)∈
    S_3(Γ_0(16),-4·),
       Φ_3( e^2π
    iz)∈ S_3(Γ_0(12),-3·),Φ_2(
    e^2π iz)∈ S_3(Γ_0(8),-8·),

where ( a·) is the Legendre-Jacobi-Kronecker symbol. See
[11,18,21]. From [21, (14.2)] we know that for odd prime p,

    α(p)= 4x^2-2p   if p=x^2+4y^2 1 4,
    
    0   if p 3 4, 4.1
    
       β(p)= 4x^2-2p   if p=x^2+2y^2 1,3 8,
    
    0   if p 5,7 8, 4.2
    
       γ(p)= 4x^2-2p   if p=x^2+3y^2 1 3,
    0   if p 2 3. 4.3

In [1] Ahlgren proved Beukers' conjecture:

    A_p-12' 4x^2-2pp^2   if p 1 4 and so p=x^2+4y^2,
    0p^2   if p>3 and p 3 4.

This is equivalent to A'_p-12α(p)p^2 for
p>3. Using modular forms with complex multiplication,  Gomez,
McCarthy and Young [11] proved that for prime p>2 and r∈
Z^+,

    A'_p^r-12 (x+yi)^2r+(x-yi)^2rp^2   if
    p=x^2+y^2 1 4 with 2∤ x,
    0p^2   if p>3 and p 3 4,
       a_p^r-12
    (x+y√(-3))^2r+(x-y√(-3))^2r p   if p=x^2+3y^2
    1 6,
    0 p   if p 5 6,
       (-1)^p^r-12f_p^r-12
       
    (x+y√(-2))^2r+(x-y√(-2))^2r p   if p=x^2+2y^2
    1,3 8,
    0 p   if p 5,7 8.

By [21, (13.1)], for odd prime p, m∈{1,3,5,…} and
r∈{2,3,4,…},

    A'_mp^r-12 (4x^2-2p)A'_mp^r-1-12-p^2 A'_mp^r-2-12p^r   if p=x^2+4y^2 1 4,
    p^2A'_mp^r-2-12p^r   if p 3 4,
       f_mp^r-12 (-1)^p-12(4x^2-2p)f_mp^r-1-12-p^2
    f_mp^r-2-12p^r   if p=x^2+2y^2 1,3 8,
    p^2f_mp^r-2-12p^r   if p 5,7 8,
       a_mp^r-12 (4x^2-2p)a_mp^r-1-12-p^2 a_mp^r-2-12p^r   if p=x^2+3y^2 1 3,
    p^2a_mp^r-2-12p^r   if p 2 3.

Now we present several conjectures, which can be viewed as generalizations
of the above results.
 Conjecture 4.3 Let p be a prime of the
form 4k+3. Then

    3p̱-̱3̱2̱p-34^2A'_p-12 p^2p^3,
    
       A'_mp^r-12 p^2A'_mp^r-2-12p^2rfor m∈{1,3,5,…} and r∈{2,3,4,…}.



Conjecture 4.4 Let p be an odd prime, m∈{1,3,5,…}
and r∈{2,3,4,…}. Then

    a_mp^r-12 p^2a_mp^r-2-12p^2r-1for p 5 6,
       f_mp^r-12 p^2f_mp^r-2-12p^2r-1for p 5,7 8,
       S_mp^r-12 p^2S_mp^r-2-12p^2r-1for p 5,7 8,
       W_mp^r-12 p^2W_mp^r-2-12p^2r-1for p 3 4,
       Q_mp^r-12 p^2Q_mp^r-2-12p^2r-1for p 13,17,19,2324.



Conjecture 4.5 Suppose that p is an odd prime,
m∈{1,3,5,…} and r∈{2,3,4,…}.
( i) If p 1,3 8 and so p=x^2+2y^2, then

    S_mp^r-12
    (4x^2-2p)S_mp^r-1-12-p^2S_mp^r-2-12p^r.

( ii) If p 1 4 and so p=x^2+4y^2, then

    W_mp^r-12
    (4x^2-2p)W_mp^r-1-12-p^2W_mp^r-2-12p^r.

( iii) If p 1,724 and so p=x^2+6y^2, then

    Q_mp^r-12
    (-1)^p-12(4x^2-2p)Q_mp^r-1-12-p^2Q_mp^r-2-12p^r.

( iv) If p 5,1124 and so p=2x^2+3y^2, then

    Q_mp^r-12
    (-1)^p+12(8x^2-2p)Q_mp^r-1-12-p^2Q_mp^r-2-12p^r.



Conjecture 4.6 Let {u_n} be one of the six sequences
{A'_n}, {f_n}, {S_n}, {a_n},{Q_n} and {W_n}, and c=-1,-8,32,9,72 or 27 according as
 u_n=A'_n,f_n,S_n,a_n,Q_n or W_n. Suppose that p is an odd prime with p∤ c.
Then

    4u_mp^2-12 (5-c^p-1)u_p-12u_mp-12p^2form=1,3,5,….

Comparing Conjecture 4.6 with the case r=2 in [21,(13.1)]
 and Conjecture 4.5 suggests the following conjecture:
 Conjecture 4.7 Suppose that p is an odd
prime.
( i) If p 1 3 and so p=x^2+3y^2, then
a_p-12 (9^p-1+3)x^2-2pp^2.
( ii) If p 1 4 and so p=x^2+4y^2, then
W_p-12 (27^p-1+3)x^2-2pp^2.
( iii) If p 1,724 and so p=x^2+6y^2, then
3pQ_p-12 (72^p-1+3)x^2-2pp^2; if p
5,1124 and so p=2x^2+3y^2, then 3pQ_p-12
(72^p-1+3)· 2x^2-2pp^2.


Remark 4.3 Let p be an odd prime. By (4.3),
 Conjecture 4.7(i) is equivalent to
 
    a_p-12(1+ 14(9^p-1-1))γ(p)p^2forp 1 3.

By (4.1),
 Conjecture 4.7(ii) is equivalent to
 
    W_p-12(1+ 14(27^p-1-1))α(p)p^2for
     p 1 4.

 By (4.2), the first congruence in [32, Conjecture 3.1] is equivalent to

    f_p-12 (-1)^p-12(1+ 14(8^p-1-1))β(p)p^2for
     p 1,3 8,

 and the first part of [31, Conjecture 2.2] is equivalent to

    S_p-12(1+ 14(32^p-1-1))β(p)p^2for
     p 1,3 8.



Let p>3 ba a prime. In [34,35] Z.W. Sun conjectured  congruences
for ∑_k=0^p-12̱ḵk^3/m^k p^2 with
m=1,-8,16,-64,256,-512,4096. Such conjectures were proved by the
author in [27], and later proved by Kibelbek et al in [15]. In [35]
Z.W. Sun also conjectured that

    ∑_k=0^p-12̱ḵk^3(-8)^k∑_k=0^p-12̱ḵk^364^k
    (-1)^p-14∑_k=0^p-12̱ḵk^3(-512)^kp^3forp 1 4,
    
       ∑_k=0^p-12̱ḵk^3-1p∑_k=0^p-12̱ḵk^34096^k∑_k=0^p-12̱ḵk^24̱ḵ2k81^kp^3forp 1,2,4 7,
    
       ∑_k=0^p-12̱ḵk^316^k-1p∑_k=0^p-12̱ḵk^3256^kp^3forp 1 3.

In light of related calculations on Maple we made the following
conjectures.
 Conjecture 4.8 Let p be a prime with p≠2,3,7.
( i) If p 1,2,4 7 and so
p=x^2+7y^2, then

    ∑_k=0^p-12̱ḵk^3
    (-1)^p-12∑_k=0^p-12̱ḵk^34096^k
    4x^2-2p-p^24x^2p^3.

( ii) If p 3,5,6 7, then

    ∑_k=0^p-12̱ḵk^33529(-1)^p-12∑_k=0^p-12̱ḵk^34096^k
       
    -114p^23̱[̱p̱/̱7̱]̱[p/7]^-2
    -11p^2[̱3̱p̱/̱7̱]̱[p/7]^-2p^3   if p 3 7,
    -9964p^23̱[̱p̱/̱7̱]̱[p/7]^-2
    -11p^2[̱6̱p̱/̱7̱]̱[2p/7]^-2p^3   if p 5 7,
    -25176p^23̱[̱p̱/̱7̱]̱[p/7]^-2
    -11p^2[̱3̱p̱/̱7̱]̱[p/7]+1^-2p^3   if p 6 7.

Remark 4.4
From [5, Theorems 9.2.6, 12.9.8 and 12.9.9] we know that if p is
an odd prime such that p 1,2,4 7 and so p=x^2+7y^2, in
1848 Eisenstein proved that

    3̱[̱ ̱p̱7̱]̱[ p7] 2x p   if p 1 7 and x 1 7,
    2x p   if p 2 7 and x 3 7,
     25x p   if p 4 7 and x 2 7.

 Conjecture 4.9 Let p>3 be
a prime.
( i) If p 1 3 and so
p=x^2+3y^2, then

    ∑_k=0^p-12̱ḵk^316^k
    (-1)^p-12∑_k=0^p-12̱ḵk^3256^k
    4x^2-2p-p^24x^2p^3.


( ii) If p 2 3, then

    ∑_k=0^p-12̱ḵk^316^k -8
    (-1)^p-12∑_k=0^p-12̱ḵk^3256^k
    -p^2p̱-̱1̱2̱p-56^-2p^3.



Conjecture 4.10 Let p be an odd prime.
( i) If p 1 4 and so
p=x^2+y^2 with 2∤ x, then

    ∑_k=0^p-12̱ḵk^3(-8)^k∑_k=0^p-12̱ḵk^364^k
    (-1)^p-14∑_k=0^p-12̱ḵk^3(-512)^k
    4x^2-2p-p^24x^2p^3.


( ii) If p 3 4, then

    ∑_k=0^p-12̱ḵk^3(-8)^k
    -3∑_k=0^p-12̱ḵk^364^k
    6(-1)^p+14∑_k=0^p-12̱ḵk^3(-512)^k
    34p^2p̱-̱3̱2̱p-34^-2p^3.



Conjecture 4.11 Let p be an odd prime. Then

    ∑_k=0^p-12̱ḵk^3(-64)^k
        (-1)^p-12(4x^2-2p-p^24x^2)p^3   if p
    1,3 8 and so p=x^2+2y^2,
    p^23[̱p̱/̱4̱]̱[p/8]^-2p^3   if p 5 8,
     32p^2[̱p̱/̱4̱]̱[p/8]^-2p^3   if p 7 8.




Conjecture 4.12 Let p be a prime with p>3. Then

    ∑_n=0^p-1A_n 4x^2-2p-p^24x^2p^3   if p
    1,3 8 and so p=x^2+2y^2,
    1727p^2[̱p̱/̱4̱]̱[p/8]^-2p^3   if p 5 8,
    -176p^2[̱p̱/̱4̱]̱[p/8]^-2p^3   if p 7 8

and

    ∑_n=0^p-1(-1)^nA_n 4x^2-2p-p^24x^2p^3   if p
    1 3 and so p=x^2+3y^2,
     54p^2p̱-̱1̱2̱p-56^-2p^3   if p 2 3.

Remark 4.5 Z.W. Sun [36] conjectured the
congruences for ∑_n=0^p-1A_n and
∑_n=0^p-1(-1)^nA_n modulo p^2 and proved the congruences
when the modulus is p. He also conjectured that

    ∑_k=0^p-1A_k∑_k=0^p-12̱ḵk^24̱ḵ2k256^kp^3forp 1,3 8,
    
       ∑_k=0^p-1(-1)^kA_k∑_k=0^p-12̱ḵk^316^kp^3forp 1 3.

Let p>3 be a prime. In [39] Z.W. Sun conjectured the congruence for
∑_n=0^p-1∑_k=0^nṉk^4 p^2, and posed
conjectures on ∑_n=0^p-1D_nm^n p^2 for
m=1,-2,4,-8,8,16,-32,64. In [30] the author proved some
congruences for ∑_n=0^p-1D_nm^n and
∑_n=0^p-1b_nm^n modulo p. Now we present
congruences for such sums modulo p^3.
 Conjecture 4.13 Let p be a prime with p≠2,3,13,47.
( i) If p
1,3 8 and so p=x^2+2y^2, then

    ∑_k=0^p-12̱ḵk^23̱ḵk8^k∑_n=0^p-1b_n
    ∑_n=0^p-1b_n81^n∑_n=0^p-1D_n8^n
    4x^2-2p-p^24x^2p^3.

( ii) If p
5,7 8, then

    ∑_k=0^p-12̱ḵk^23̱ḵk8^k    -3347∑_n=0^p-1b_n
     -9913∑_n=0^p-1b_n81^n
    33∑_n=0^p-1D_n8^n
       119p^2[̱
    ̱p̱/̱4̱]̱[p/8]^-2p^3   if p 5 8,
    -112p^2[̱p̱/̱4̱]̱[p/8]^-2p^3   if p 7 8.



Conjecture 4.14 Let p be a prime with p≠2,3,5,13. If
p 1,17,19,2330, then

    ∑_k=0^p-12̱ḵk^23̱ḵk(-27)^k∑_k=0^p-12̱ḵk^23̱ḵk15^3k
    p3∑_n=0^p-1∑_k=0^nṉk^4 ∑_n=0^p-1D_n∑_n=0^p-1D_n64^n
        4x^2-2p-p^24x^2p^3   if p
    1,1930 and so p=x^2+15y^2,
    2p-12x^2+p^212x^2p^3   if p
    17,2330 and so p=3x^2+5y^2.

If p 7,11,13,2930, then

    ∑_k=0^p-12̱ḵk^23̱ḵk(-27)^k    28∑_k=0^p-12̱ḵk^23̱ḵk15^3k
    p3∑_n=0^p-1∑_k=0^nṉk^4 2853∑_n=0^p-1D_n
    -11213∑_n=0^p-1D_n64^n
        72p^2· 5^[p/3][̱p̱/̱3̱]̱[p/15]^-2p^3   if p
    730,
    14p^2· 5^[p/3][̱p̱/̱3̱]̱[p/15]^-2p^3   if p
    1130,
     732p^2·
    5^[p/3][̱p̱/̱3̱]̱[p/15]^-2p^3   if p 1330,
     78p^2· 5^[p/3][̱p̱/̱3̱]̱[p/15]^-2p^3   if p
    2930.



Conjecture 4.15 Let p>3 be a prime.
( i) If p
1 3 and so p=x^2+3y^2, then

    ∑_k=0^p-12̱ḵk^23̱ḵk108^k   ∑_n=0^p-12̱ṉnf_n(-4)^n∑_k=0^p-12̱ḵk^23̱ḵk1458^k∑_n=0^p-1D_n(-2)^n∑_n=0^p-1D_n4^n∑_n=0^p-1D_n16^n
       ∑_n=0^p-1D_n(-32)^n∑_n=0^p-1b_n(-9)^n 4x^2-2p-p^24x^2p^3.

( ii) If p
2 3, then

    ∑_k=0^p-12̱ḵk^23̱ḵk108^k   -∑_n=0^p-12̱ṉnf_n(-4)^n -
    14∑_n=0^p-1D_n(-2)^n -∑_n=0^p-1D_n4^n
    2∑_n=0^p-1D_n16^n
       
    -∑_n=0^p-1D_n(-32)^n -2∑_n=0^p-1b_n(-9)^n -p^22p̱-̱1̱2̱p-56^-2p^3.



Conjecture 4.16 Let p>3 be a prime.
( i) If p 1 4, then

    ∑_n=0^p-1b_n(-3)^n∑_n=0^p-1b_n(-27)^n∑_k=0^p-12̱ḵk^24̱ḵ2k(-12288)^k
       
    4x^2-2p-p^24x^2p^3   if p 112 and so
    p=x^2+9y^2,
    2p-2x^2+p^22x^2p^3   if p 512 and so
    2p=x^2+9y^2.

( ii) If p 3 4, then

    ∑_n=0^p-1b_n(-3)^n    -15
    ∑_n=0^p-1b_n(-27)^n 10∑_k=0^p-12̱ḵk^24̱ḵ2k(-12288)^k
        -
    53p^2[̱p̱/̱3̱]̱[p/12]^-2p^3   if p 712,
     56p^2[̱p̱/̱3̱]̱[p/12]^-2p^3   if p
    1112.



 Conjecture 4.17 Let
p be a prime with p>7.
( i) If p 1,2,4 7 and so p=x^2+7y^2, then

    p3∑_n=0^p-12̱ṉnW_n(-27)^n   ∑_k=0^p-12̱ḵk^24̱ḵ2k81^k∑_k=0^p-12̱ḵk^24̱ḵ2k(-3969)^k
       -15p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-15)^3k
    4x^2-2p-p^24x^2p^3.

( ii) If p 3,5,6 7, then

    p3∑_n=0^p-12̱ṉnW_n(-27)^n - 940∑_k=0^p-12̱ḵk^24̱ḵ2k81^k4528∑_k=0^p-12̱ḵk^24̱ḵ2k(-3969)^k
        -375752-15p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-15)^3k 516p^23̱[̱p̱/̱7̱]̱[p/7]^-2p^3   if p
    3 7,
    45256p^23̱[̱p̱/̱7̱]̱[p/7]^-2p^3   if p 5 7,
    1257744p^23̱[̱p̱/̱7̱]̱[p/7]^-2p^3   if p 6 7.




Conjecture 4.18 Let p be a prime with p>7 and p≠71.
( i) If p 1,3 8 and
so p=x^2+2y^2, then

    ∑_k=0^p-12̱ḵk^24̱ḵ2k256^k   ∑_k=0^p-12̱ḵk^24̱ḵ2k28^4k-5p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k20^3k
    4x^2-2p-p^24x^2p^3.

( ii) If p 5,7 8, then

    ∑_k=0^p-12̱ḵk^24̱ḵ2k256^k    -44171∑_k=0^p-12̱ḵk^24̱ḵ2k28^4k
    -257-5p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k20^3k
       p^23[̱p̱/̱4̱]̱[p/8]^-2p^3   if p 5 8,
    - 32p^2[̱p̱/̱4̱]̱[p/8]^-2p^3   if p 7 8.



Conjecture 4.19 Let p be a prime with p>5.
( i) If p 1 3 and so p=x^2+3y^2, then

    ∑_k=0^p-12̱ḵk^24̱ḵ2k(-144)^k p5∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k54000^k
    4x^2-2p-p^24x^2p^3.

( ii) If p 2 3, then

    ∑_k=0^p-12̱ḵk^24̱ḵ2k(-144)^k 25  p5∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k54000^k
    p^2(̱p̱-̱1̱)̱/̱2̱(p-5)/6^-2p^3.



Conjecture 4.20 Let p be a prime with p≠2,3,11.
( i) If p 1 4 and so
p=x^2+y^2 with 2∤ x,

    p3∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k12^3k   33p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k66^3k∑_k=0^p-12̱ḵk^24̱ḵ2k648^k
       
    p3∑_k=0^p-12̱ḵkW_k54^k 4x^2-2p-p^24x^2p^3.

( ii) If p 3 4, then

    p3∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k12^3k   1211333p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k66^3k
    -3∑_k=0^p-12̱ḵk^24̱ḵ2k648^k
        - 53
    p3∑_k=0^p-12̱ḵkW_k54^k
    512p^2(̱p̱-̱3̱)̱/̱2̱(p-3)/4^-2p^3.



Conjecture 4.21 Let p be a prime with p>5.
( i) If p 1 3 and so 4p=L^2+27M^2, then

    ∑_k=0^p-12̱ḵk^23̱ḵk(-192)^k10p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-12288000)^k L^2-2p-p^2L^2p^3.

( ii) If p 2 3, then

    ∑_k=0^p-12̱ḵk^23̱ḵk(-192)^k   80016110p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-12288000)^k
         3p^2[̱2̱p̱/̱3̱]̱[p/12]^-2 34p^2[̱2̱p̱/̱3̱]̱[p/3]^-2p^3   if 12| p-5,
     349p^2[̱2̱p̱/̱3̱]̱[p/12]^-2 34p^2[̱2̱p̱/̱3̱]̱[p/3]^-2p^3   if 12| p-11.



Conjecture 4.22 Let p be a prime with p≠2,11,13.
( i) If p 1,3,4,5,911 and so 4p=u^2+11v^2
with u,v∈ Z, then

    ∑_k=0^p-12̱ḵk^23̱ḵk64^k-2p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-32)^3k
    u^2-2p-p^2u^2p^3.

( ii) If p 2,6,7,8,1011  and
f=[ p11], then

    ∑_k=0^p-12̱ḵk^23̱ḵk64^k   16039-2p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-32)^3k
        -p^254̱f̱2f23̱f̱f6̱f̱3f^2p^3   if
    p 211,
    -p^2134̱f̱2f303̱f̱f6̱f̱3f^2p^3   if p 611,
    -p^2854̱f̱2f5583̱f̱f6̱f̱3f^2p^3   if p
    711,
    -p^274̱f̱2f1483̱f̱f6̱f̱3f^2p^3   if p
    811,
    -p^2294̱f̱2f7563̱f̱f6̱f̱3f^2p^3   if p
    1011.




Conjecture 4.23 Let p be a prime with p≠2,3,19.
( i) If p 1,4,5,6,7,9,11,16,1719 and so 4p=u^2+19v^2
with u,v∈ Z, then

    -6p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-96)^3k
    u^2-2p-p^2u^2p^3.

( ii) If p 2,319  and
f=[ p19], then

    -6p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3k(-96)^3k
    -985384p^26̱f̱3f1̱0̱f̱2f6̱f̱f1̱0̱f̱3f1̱0̱f̱4f^2
    p^3   if 19| p-2,
    -19758080p^26̱f̱3f1̱0̱f̱2f6̱f̱f1̱0̱f̱3f1̱0̱f̱4f^2
    p^3   if 19| p-3.



Conjecture 4.24 Let p be a prime with p≠2,3,23.
( i) If p 1,5,7,1124, then

    ∑_k=0^p-12̱ḵk^23̱ḵk216^k p3∑_k=0^p-12̱ḵk^24̱ḵ2k48^2k
    p3∑_n=0^p-1b_n9^n∑_n=0^p-1D_n(-8)^n
        4x^2-2p-p^24x^2p^3   if p 1,724 and
    so p=x^2+6y^2,
    8x^2-2p-p^28x^2p^3   if p 5,1124 and so p=2x^2+3y^2.

( ii) If p 13,17,19,2324, then

    ∑_k=0^p-12̱ḵk^23̱ḵk216^k -7 p3∑_k=0^p-12̱ḵk^24̱ḵ2k48^2k
    723 p3∑_n=0^p-1b_n9^n - 725∑_n=0^p-1D_n(-8)^np^3.

Remark 4.6 Suppose that p is a prime such that
p 1,5,7,1124. In [35] Z.W. Sun conjectured that
∑_k=0^p-12̱ḵk^23̱ḵk/216^k 
p3∑_k=0^p-12̱ḵk^24̱ḵ2k/48^2kp^3.

Conjecture 4.25 Let p be a prime such that p>3 and p
3,5,6 7. Then

    p3∑_k=0^p-12̱ḵk4̱ḵ2k63^k p33̱[̱p̱/̱7̱]̱[p/7]p^2   if p 3 7,
    - p43̱[̱p̱/̱7̱]̱[p/7]p^2   if p 5 7,
    5p663̱[̱p̱/̱7̱]̱[p/7]p^2   if p 6 7.

Remark 4.7 For the conjecture for ∑_k=0^p-12̱ḵk4̱ḵ2k/63^k p^2 in the cases p 1,2,4 7
see [35, Conjecture 5.14].
 In [27] the author showed that these
congruences are true when the modulus is p. For any prime p>3,
in [35] Z.W. Sun also made conjectures on
∑_k=0^p-12̱ḵk4̱ḵ2k/48^k and
∑_k=0^p-12̱ḵk4̱ḵ2k/72^k modulo p^2.
Conjecture 4.26 Let p be a prime with p 1 3. Then

    ∑_k=1^p-1kW_k(-9)^k 0p^2.



Conjecture 4.27 Let p be a prime with p>3. Then

    ∑_k=0^p-12̱ḵkW_k(-12)^k
        L^2-2pp^2   if p 1 3 and so 4p=L^2+27M^2 with L,M∈ Z,
    0p^2   if p 2 3.



Conjecture 4.28 Let p be an odd prime,
 n∈{-640320,-5280,-960,-96,-32, 20,255} and
n(n-12)0 p. Then

    ∑_k=0^p-12̱ḵkW_k(n-12)^kn(n-12)p∑_k=0^p-12̱ḵk3̱ḵk6̱ḵ3kn^3kp^2.
 Hence the congruences in Theorems 2.4-2.10
also hold modulo p^2.


Conjecture 4.29 Let p>3 be a prime. Then

    ∑_k=0^p-12̱ḵk3k+1(-16)^kf_k (-1)^p-12p
    +p^3E_p-3p^4,
    
       ∑_k=0^p-12̱ḵk7k+2(-27)^kW_k 2-3p p
    -4p^3U_p-3p^4,
    
       ∑_k=0^p-12̱ḵk7k+38^kW_k 3-2ppp^2,
    
       ∑_k=0^p-12̱ḵk7k+254^kW_k 2-3ppp^2,
    
       ∑_k=0^p-12̱ḵk14k+3(-44)^kW_k 3-11ppp^2forp≠11,
    
       ∑_k=0^p-12̱ḵk38k+7(-108)^kW_k 7-3ppp^2,
    
       ∑_k=0^p-12̱ḵk133k+26243^kW_k 26-3ppp^2,
    
       ∑_k=0^p-12̱ḵk602k+85(-972)^kW_k 85-3ppp^2,
    
       ∑_k=0^p-12̱ḵk4154k+481(-5292)^kW_k 481-3ppp^2for p≠7.

Remark 4.8 In [12] Guo proved that for any odd prime p,

    ∑_k=0^p-12̱ḵk3k+1(-16)^kf_k (-1)^p-12pp^3.



99
[1]  S. Ahlgren, Gaussian hypergeometric series and combinatorial congruences, in:
Symbolic computation, number theory, special functions, physics and
combinatorics (Gainesville, FI, 1999), pp.1-12, Dev. Math., Vol. 4,
Kluwer, Dordrecht, 2001, pp.1-12.

[2]  S. Ahlgren and K. Ono, A Gaussian hypergeometic series evaluation
and Apéry number
 congruences, J. Reine Angew. Math. 518(2000), 187-212.

[3]  T. Amdeberhan and R. Tauraso, Supercongruences
for the Almkvist-Zudilin numbers, Acta Arith. 173(2016),
255-268.

[4]  R. Apéry, Irrationalité de ζ(2) et ζ(3),
 Astérisque 61(1979), 11-13.

 [5]  B.C. Berndt, R.J. Evans and K.S. Williams, Gauss and Jacobi Sums, Wiley, New York, 1998.

[6]  F. Beukers, Some congruences for the Apéry numbers,
 J. Number Theory 21(1985), 141-155.

[7]  F. Beukers, Another congruence for the Apéry numbers,
 J. Number Theory 25(1987), 201-210.

[8] H.H. Chan, S. Cooper and F. Sica, Congruences
satisfied by Apéry-like numbers, Int. J. Number Theory 6(2010), 89-97.

[9] H.H. Chan, Y. Tanigawa, Y. Yang and W. Zudilin,
 New analogues of Clausen's identities arising from the theory
 of modular forms, Adv. Math. 228(2011), 1294-1314.

 [10]  H.H. Chan and W. Zudilin, New representations
 for Apéry-like sequences, Mathematika 56(2010), 107-117.

[11]  A. Gomez, D. McCarthy, D. Young, Apéry-like numbers and
families of newforms with complex multiplication, Res. Number
Theory 2019, 5:5, pp.1-12.

[12]  V.J.W. Guo, Proof of a supercongruence
conjectured by Z.-H. Sun, Integral Transforms Spec. Funct. 25
(2014), 1009-1015.

[13]  F. Jarvis and H.A. Verrill,
Supercongruences for the Catalan-Larcombe-French numbers,
Ramanujan J. 22(2010), 171-186.

[14]  X.J. Ji and Z.H. Sun, Congruences for
Catalan-Larcombe-French numbers, Publ. Math. Debrecen 90(2017), 387-406.

[15]  J. Kibelbek, L. Long, K. Moss, B. Sheller and H. Yuan,
Supercongruences and complex multiplication, J. Number Theory
164(2016), 166-178.

[16]  J.-C. Liu, Supercongruences involving p-adic
Gamma functions, Bull. Aust. Math. Soc. 98(2018), 27-37.

[17]  A. Malik and A. Straub, Divisibility
properties of sporadic Apéry-like numbers, Res. Number Theory
2016, 2:5, pp.1-26.

[18]  D. McCarthy, R. Osburn, A. Straub, Sequences, modular forms and
cellular integrals, Math. Proc. Cambridge Philos. Soc.,
doi:10.1017/S0305004118000774.

[19]  E. Mortenson, Supercongruences for truncated _n+1F_n hypergeometric series with applications to certain weight
three newforms, Proc. Amer. Math. Soc.133(2005), 321-330.

[20]  S. Ramanujan, Modular equations and approximations to π, Quart. J.
Math. (Oxford) 45(1914), 350-372.

[21]  J. Stienstra and F. Beukers, On the
Picard-Fuchs equation and the formed Brauer group of certain
elliptic K3-surfaces, Math. Ann. 271(1985), 269-304.

[22]  Z.H. Sun, Congruences concerning Bernoulli
numbers and Bernoulli polynomials, Discrete Appl. Math. 105(2000), 193-223.

[23]  Z.H. Sun, On the number of incongruent
residues of x^4+ax^2+bx modulo p, J. Number Theory 119(2006), 210-241.

[24]  Z.H. Sun, Congruences involving Bernoulli and
Euler numbers, J. Number Theory 128(2008), 280-312.

[25] 
Z. H. Sun, Congruences concerning Legendre polynomials, Proc.
Amer. Math. Soc. 139(2011), 1915-1929.

[26]  Z.H. Sun, Identities and congruences for a new sequence, Int. J.
Number Theory 8(2012), 207-225.

[27] 
Z. H. Sun, Congruences concerning Legendre polynomials II, J.
Number Theory 133(2013), 1950-1976.


[28] 
Z. H. Sun, Congruences involving 2̱ḵk^23̱ḵk, J. Number
Theory 133(2013), 1572-1595.


[29] 
Z. H. Sun, Legendre polynomials and supercongruences, Acta
Arith. 159(2013), 169-200.


[30]  Z. H. Sun, Congruences for Domb and
Almkvist-Zudilin numbers, Integral Transforms Spec. Funct. 26(2015), 642-659.

[31]  Z.H. Sun, Identities and congruences for
Catalan-Larcombe-French numbers, Int. J. Number Theory 13(2017), 835-851.

[32]  Z.H. Sun, Congruences for sums involving
Franel numbers, Int. J. Number Theory 14(2018), 123-142.

[33]  Z.H. Sun, Super congruences for two
Apéry-like sequences, J. Difference Equ. Appl. 24(2018),
1685-1713.

[34]  Z.W. Sun, On congruences related to central binomial
 coefficients,
J. Number Theory 131(2011), 2219-2238.

[35]  Z.W. Sun, Super congruences and Euler
numbers, Sci. China Math. 54(2011), 2509-2535.


[36]  Z.W. Sun, On sums of Apery polynomials and
related congruences, J. Number Theory 132(2012), 2673-2699.

[37]  Z.W. Sun, Connections between p=x^2+3y^2 and
Franel numbers, J. Number Theory 133(2013), 2914-2928.

[38]   Z.W. Sun, Congruences for Franel numbers,
Adv. in Appl. Math. 51(2013), 524-535.


[39]  Z.W. Sun, Conjectures and results on x^2 mod p^2
with 4p=x^2+dy^2, In: Number Theory and Related Areas,
International Press of Boston, Somerville, MA, 2013, 149-197.

[40]  The On-Line Encyclopedia of Integer
Sequences, http://oeis.org.

[41] D. Zagier,
Integral solutions of Apéry-like recurrence equations, In:
Groups and Symmetries, American  Mathematical Society, Providence,
RI, 2009, 349-366.



    

Zhi-Hong SunSchool of Mathematics and
Statistics
Huaiyin Normal University
 Huaian, Jiangsu 223300
 P.R. China
Email: zhsun@hytc.edu.cn
 URL: http://maths.hytc.edu.cn/szh1.htm

    

 
