Congruences for generalized Apéry numbers and Gaussian hypergeometric series

Abstract

For positive integers $f_1,f_2,m,l$, we define a generalization of Apéry numbers $A(f_1,f_2,m,l,\lambda )$ given by

$$\begin{aligned} A(f_1,f_2,m,l,\lambda )\,{:=}\,\sum _{j=0}^{f_2}{f_1+j\atopwithdelims ()j}^m{f_2\atopwithdelims ()j}^{l}\lambda ^j. \end{aligned}$$

In this article, we deduce congruence relations satisfied by these generalized Apéry numbers extending results of (Coster in Supercongruences, Ph.D. thesis, Universiteit Leiden, 1988). We find expressions of $A(f_1,f_2,m,l,\lambda )$ in terms of Gaussian hypergeometric series and evaluate some new supercongruences similar to Beukers’ supercongruences.

1 Introduction and statement of results

In [3], R. Apéry introduced the numbers $a_n$ and $b_n$ commonly known as Apéry numbers defined by the explicit formulas

$$\begin{aligned} a_n\,{:=}\,\sum _{k=0}^{n}{n+j\atopwithdelims ()j}^2{n\atopwithdelims ()j}^2~~~\text {and}~~~b_n\,{:=}\,\sum _{k=0}^{n}{n+j\atopwithdelims ()j}{n\atopwithdelims ()j}^2 \end{aligned}$$

in his irrationality proof of $\zeta (3)$ and $\zeta (2)$, respectively. Since their appearance, mathematicians started looking more closely at these numbers and found many interesting properties satisfied by them. Chowla et al. [11] were first to consider congruence properties for the Apéry numbers. They proved many interesting congruence relations for the Apéry numbers and listed a few conjectures relating them. For example, they proved that for primes $p\ge 5$,

$$\begin{aligned} a_p\equiv a_1~(\text {mod}~p^3). \end{aligned}$$

Gessel [13] generalized this to the stronger result

$$\begin{aligned} a_{np}\equiv a_n~(\text {mod}~ p^3). \end{aligned}$$

(1.1)

In a recent paper [10], Chan, Cooper, and Sica investigated certain sequences of integers $\{f_n\}_{n=1}^\infty$ that satisfy relations similar to (1.1). The identification of $a_n$ as the coefficients of certain power series motivates them to obtain Apéry-like sequences $\{f_n\}_{n=1}^{\infty }$ satisfying congruences similar to (1.1).

One main purpose of this paper is to find similar congruences for the numbers $A(f_1,f_2,m,l,\lambda )$ given by the formula

$$\begin{aligned} A(f_1,f_2,m,l,\lambda )\,{:=}\,\sum _{j=0}^{f_2}{f_1+j\atopwithdelims ()j}^m{f_2\atopwithdelims ()j}^{l}\lambda ^j, \end{aligned}$$

where $f_1,f_2,m,l\in \mathbb {N}$. Note that this family of sequences includes the Apéry numbers. In his PhD thesis [12], Coster studied the numbers $A(f_1,f_1,l,m,\lambda )$ and proved that

$$\begin{aligned} A(f_1p^r,f_1p^r,m,l,\lambda )\equiv A(f_1p^{r-1},f_1p^{r-1},m,l,\lambda )~~(\text {mod} ~p^{3r}) \end{aligned}$$

if $p\ge 5$ and $\lambda =\pm 1$. Our main result extends the result of Coster and proves the following congruence for the numbers $A(f_1,f_2,m,l,\pm 1)$.

Theorem 1.1

Let $\lambda =\pm 1$. For primes $p>3$ and integers $l\ge 2$, $r\ge 1$, we have

$$\begin{aligned} A(f_1p^r,f_2p^r,m,l,\lambda )\equiv A(f_1p^{r-1},f_2p^{r-1},m,l,\lambda )~~({\mathrm{mod}}~p^{3r}). \end{aligned}$$

Let p be an odd prime, and $\mathbb {F}_p$ denote the finite field with p elements. A multiplicative character $\chi :\mathbb {F}_p^\times \rightarrow \mathbb {C}$ is a group homomorphism. The set $\widehat{\mathbb {F}_p^\times }$ of all multiplicative characters on $\mathbb {F}_p^\times$ forms a cyclic group under multiplication of characters. Extend each character $\chi \in \widehat{\mathbb {F}_p^{\times }}$ to all of $\mathbb {F}_p$ by setting $\chi (0)\,{:=}\,0$. The binomial coefficient ${A \atopwithdelims ()B}$ is defined by

$$\begin{aligned} {A \atopwithdelims ()B}\,{:=}\,\frac{B(-1)}{q}J(A,\overline{B})=\frac{B(-1)}{p}\sum _{x \in \mathbb {F}_p}A(x)\overline{B}(1-x), \end{aligned}$$

where J(A, B) denotes the usual Jacobi sum and $\overline{B}$ is the inverse of B. The following property of binomial coefficients is known from [14]

$$\begin{aligned} \left( {\begin{array}{c}A\\ B\end{array}}\right) =\left( {\begin{array}{c}B\overline{A}\\ B\end{array}}\right) B(-1). \end{aligned}$$

(1.2)

With this notation, for characters $A_0, A_1,\ldots , A_r$ and $B_1, B_2,\ldots , B_r$ of $\mathbb {F}_p$, Greene [14] defined the Gaussian hypergeometric series ${_{r+1}}F_r\left( \begin{array}{cccc} A_0, &{} A_1, &{} \ldots , &{} A_r\\ &{} B_1, &{} \ldots , &{} B_r \end{array}\mid x \right)$ over $\mathbb {F}_p$ as

$$\begin{aligned} {_{r+1}}F_r\left( \begin{array}{cccc} A_0, &{} A_1, &{} \ldots , &{} A_r\\ &{} B_1, &{} \ldots , &{} B_r \end{array}\mid x \right) \,{:=}\,\frac{p}{p-1}\sum _{\chi }{A_0\chi \atopwithdelims ()\chi }{A_1\chi \atopwithdelims ()B_1\chi } \cdots {A_r\chi \atopwithdelims ()B_r\chi }\chi (x), \end{aligned}$$

where the sum is over all characters $\chi$ of $\mathbb {F}_p$.

For an odd prime p, Beukers [9] and Stienstra-Beukers [25] proved many interesting congruence properties satisfied by Apéry numbers, and they respectively made the following two conjectures concerning the Apéry numbers and the coefficients of two modular forms:

$$\begin{aligned} a_{\frac{p-1}{2}}\equiv \alpha (p)~(\text {mod}~p^2),\end{aligned}$$

(1.3)

$$\begin{aligned} b_{\frac{p-1}{2}}\equiv \beta (p)~(\text {mod}~p^2), \end{aligned}$$

(1.4)

where

$$\begin{aligned} \sum _{n=0}^{\infty }\alpha (n)q^n&~\,{:=}\,q\prod _{n=1}^{\infty }(1-q^{2n})^4(1-q^{4n})^4,\\ \sum _{n=0}^{\infty }\beta (n)q^n&~\,{:=}\,q\prod _{n=1}^{\infty }(1-q^{4n})^6. \end{aligned}$$

For $p\not \mid a_{\frac{p-1}{2}}$, Ishikawa [17] gave a proof (1.3). In [1], Ahlgren and Ono proved (1.3) by relating the Gaussian hypergeometric series ${_{4}}F_3(1)$ to $\alpha (p)$ via the modularity of a certain Calabi-Yau threefold. Moreover, they defined the generalized Apéry numbers A(n, n, l, k, 1) and found certain Beukers-like congruences for these generalized Apéry numbers exploring connections between Gaussian hypergeometric series, elliptic curves and the Dedekind eta function. Following the technique of [1], Ahlgren [2] studied ${_{3}}F_2(\lambda )$ Gaussian hypergeometric series using tools from p-adic analysis and deduced (1.4), which was already proved by vanHamme [26] for $p\equiv 1$ (mod 4) and by Ishikawa [16] in general. Further, he evaluated some new supercongruences similar to (1.4) by inserting Beukers’ supercongruences into a larger framework.

One of our main aims in this paper is to investigate similar phenomena for the generalized Apéry numbers $A(m,n,l,k,\lambda )$ and Gaussian hypergeometric series ${_{r+1}}F_r(\lambda )$. In [20], Koike found an expression of $A\left( \frac{p-1}{2},\frac{p-1}{2},m,l,1\right)$ in terms of Gaussian hypergeometric series with quadratic and trivial characters as parameters. Following this, Ono [22] deduced a similar congruence relation for $A\left( \frac{p-1}{2},\frac{p-1}{2},m,l,\lambda \right)$. In the following theorem, we extend these results of [20, 22] and evaluate an expression of $A\left( \frac{p-1}{r},\frac{p-1}{s},m,l,\lambda \right)$ in terms of Gaussian hypergeometric series.

Theorem 1.2

Let p be an odd prime such that $p\equiv 1$ (mod lcm(r, s)). If $w=m+l$ and $0<a<r$, $0<b<s$, then

$$\begin{aligned}&A\left( \frac{a(p-1)}{r},\frac{b(p-1)}{s},m,l,\lambda \right) \\&\quad \equiv (-p)^{w-1}{_w}F_{w-1}\left( \begin{array}{cccccccc} \psi ^a, &{} \psi ^a, &{}\ldots , &{}\psi ^a, &{}\overline{\xi }^b, &{}\overline{\xi }^b, &{} \ldots , &{} \overline{\xi }^b\\ &{} \epsilon ,&{} \ldots , &{} \epsilon , &{} \epsilon , &{} \epsilon , &{}\cdots , &{} \epsilon \end{array} \mid (-1)^{l}\lambda \right) ~~({\mathrm{mod}} ~p), \end{aligned}$$

where the character $\psi ^a$ of order r appears m times and the character $\xi ^b$ of order s appears l times.

As a consequence of the above theorem, we evaluate the following congruences for the generalized Apéry numbers.

Theorem 1.3

Let $p\equiv 1$ (mod 4) be a prime and $d=\frac{p-1}{4}$, then

$$\begin{aligned} (i) \,\,A\left( d,d,1,1,-\frac{4}{3}\right)&\equiv \chi _4(-1)A\left( d,d,1,1,\frac{1}{3}\right) \\&\equiv \left\{ \begin{array}{ll} 0~({\mathrm{mod}}~ p), &{}\quad \hbox {if}\,p\equiv 2 (\mathrm{mod} \,3); \\ \displaystyle -2a_3\chi _4(-36)~({\mathrm{mod}}~ p), &{}\quad \hbox {if}\,p\equiv 1 (\mathrm{mod} \,3), \end{array} \right. \\ (ii) \,\,A(d,d,1,1,-4)&\equiv \chi _4(4)A\left( d,d^3,1,1,-\frac{1}{4}\right) \\&\equiv \left\{ \begin{array}{ll} 0~({\mathrm{mod}}~ p), &{} \quad \hbox {if}\,p\equiv 2 (\mathrm{mod} \,3); \\ \displaystyle -2a_3\chi _4(12)~({\mathrm{mod}}~ p),&{} \quad \hbox {if}\,p\equiv 1 (\mathrm{mod} \,3), \end{array} \right. \end{aligned}$$

where $p=a_3^2+3b_3^2$ and $a_3\equiv -1$ (mod 3), when $p\equiv 1$ (mod 3).

We now move our attention to supercongruence relations satisfied by the generalized Apéry numbers $A\left( \frac{r-1}{r}(p-1),\frac{p-1}{r},m,m(r-1),\lambda \right)$. For odd prime p, we define

$$\begin{aligned} B\left( f_1,f_2,m,l,\lambda \right) :=\sum _{j=0}^{f_2}\left( {\begin{array}{c}f_{1}+j\\ j\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2}\\ j\end{array}}\right) ^{l}\lambda ^j\left\{ 1+\frac{m(p-1)}{f_2}j\left\{ H_{\frac{pf_2-f_1}{p-1}+j}-H_j\right\} \right\} , \end{aligned}$$

where $H_0\,{:=}\,0$ and $H_n\,{:=}\,1+\frac{1}{2}+\ldots +\frac{1}{n}$ for $n\in \mathbb {N}$. In the following theorem, we express Gaussian hypergeometric series in terms of the quantities $A\left( f_1,f_2,m,l,\lambda \right)$ and $B\left( f_1,f_2,m,l,\lambda \right)$.

Theorem 1.4

Let $p\equiv 1$ (mod r) and $T\in \widehat{\mathbb {F}_q^\times }$ be a generator of the character group, then

$$\begin{aligned}&(-p)^{mr-1}{_{mr}}F_{mr-1} \left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{m(r-1)}\lambda \right) \\&\quad \equiv \,A\left( \frac{r-1}{r}(p-1),\frac{p-1}{r},m,m(r-1),\lambda ^p\right) \\&\qquad +pB\left( \frac{r-1}{r}(p-1),\frac{p-1}{r},m,m(r-1),\lambda \right) ~~({\mathrm{mod}}~p^2). \end{aligned}$$

Remark 1.5

In [1], Ahlgren and Ono obtained a particular case of Theorem 1.4 with $r=2,m=2$. Moreover, they deduced the Beukers’ conjecture (1.3) by proving the combinatorial identity $B\left( \frac{p-1}{2},\frac{p-1}{2},2,2,1\right) \equiv 0~~(\text {mod}~p)$.

Remark 1.6

For $r=2$ and $m\in \mathbb {N}$, Osburn and Schneider [23] found a modulo $p^3$ version of Theorem 1.4 extending results of [1, 2]. In particular, they deduced a supercongruence for the Legendre symbol $\left( \frac{-1}{p}\right)$, which generalized a supercongruence conjecture of Rodriguez-Villegas proved by Mortenson [21].

For $\lambda \in \mathbb {Q}\backslash \{0,1\}$, consider the family of elliptic curves

$$\begin{aligned} {_{2}}E_1(\lambda ):y^2=x(x-1)(x-\lambda ) \end{aligned}$$

and denote by ${_{2}}a_1(p, \lambda )$ the trace of the Frobenius endomorphism of ${_{2}}E_1(\lambda )$ over $\mathbb {F}_p$. Then the associated Hasse-Weil L-function of ${_{2}}F_1(\lambda )$ is given by $L({_{2}}E_1(\lambda ),s)=\sum _{n=1}^\infty \frac{{_{2}}a_1(n,\lambda )}{n^s}$. Koike [20] proved that if p is an odd prime for which ord$_p(\lambda (\lambda -1))=0$, then

$$\begin{aligned} {_{2}}a_1(p, \lambda )=-p\phi (-1){_{2}}F_1\left( \begin{array}{cccc} \phi , &{} \phi \\ &{} \epsilon \end{array}\mid \lambda \right) . \end{aligned}$$

This, together with Theorem 1.4 yields

Corollary 1.7

If p is an odd prime and $\lambda \in \mathbb {Q}\backslash \{0,1\}$ such that ord$_p(\lambda (\lambda -1))=0$, then

$$\begin{aligned} {_{2}}a_1(p,\lambda )&\equiv \phi (-1)A\left( \frac{p-1}{2},\frac{p-1}{2},1,1,-\lambda ^p\right) \\&\quad +\phi (-1)pB\left( \frac{p-1}{2},\frac{p-1}{2},1,1,-\lambda \right) ~~({\mathrm{mod}}~p^2). \end{aligned}$$

A particular case of [5, Theorem. 3.2] states that if $p\equiv 1$ (mod 4), then the trace of Frobenius map on the elliptic curve $E(\lambda )\,{:}\, y^2=x^3+\lambda x^2+x$ over $\mathbb {F}_p$ is given by

$$\begin{aligned} a(p, \lambda )=-p\phi (2\lambda )\chi _4(-1){_{2}}F_1\left( \begin{array}{cccc} \chi _4, &{} \chi _4^3\\ &{} \epsilon \end{array}\mid \frac{4}{\lambda ^2} \right) . \end{aligned}$$

With a change of variables in [18, (4.2)], we obtain

$$\begin{aligned} {_{2}}F_1\left( \begin{array}{cccc} \phi , &{} \phi \\ &{} \varepsilon \end{array}\mid \frac{4}{2-\lambda } \right) =\phi (2\lambda (\lambda -2))\chi _4(-1){_{2}}F_1\left( \begin{array}{cccc} \chi _4, &{} \chi _4^3\\ &{} \varepsilon \end{array}\mid \frac{4}{\lambda ^2} \right) . \end{aligned}$$

As a result, we deduce

$$\begin{aligned} a(p, \lambda )=-p\phi (\lambda -2){_{2}}F_1\left( \begin{array}{cccc} \phi , &{} \phi \\ &{} \epsilon \end{array}\mid \frac{4}{2-\lambda } \right) . \end{aligned}$$

Therefore, we have the following immediate consequence of Theorem 1.4.

Corollary 1.8

Let p be an odd prime and $\lambda \in \mathbb {Q}^\times$ such that ord$_p(\lambda ^2-4)=0$, then

$$\begin{aligned} a(p,\lambda )&\equiv \phi (\lambda -2)A\left( \frac{p-1}{2},\frac{p-1}{2},1,1,\left( \frac{4}{\lambda -2}\right) ^p\right) \\&\quad +\phi (\lambda -2)pB\left( \frac{p-1}{2},\frac{p-1}{2},1,1,\frac{4}{\lambda -2}\right) ~~({\mathrm{mod}}~p^2). \end{aligned}$$

The organization of this paper is as follows. In Section 2, we prove Theorem 1.1 using the techniques of [13] and [24]. We prove Theorem 1.2 in Section 3, and deduce the congruence relations for $A(\frac{p-1}{r},\frac{p-1}{s},m,l,\lambda )$ stated in Theorem 1.3. In Section 4, we recall necessary properties of Jacobi sums and p-adic Gamma function required to prove Theorem 1.4 in Section 5. Finally, in the last section we give examples of certain supercongruences similar to Beukers’ supercongruences as given in [2, Section 5].

2 Proof of Theorem 1.1

We recall the following result from [24].

Lemma 2.1

[24, Lemma 2.2] Let p be a prime and n an integer such that $n\not \equiv 0$ (mod $p-1)$. Then, for all integers $r\ge 0$,

$$\begin{aligned} \sum _{k=1; p\not \mid k}^{p^r-1} k^n\equiv 0~~({\mathrm{mod}}~p^r). \end{aligned}$$

If, additionally, n is even, then, for primes $p\ge 5$,

$$\begin{aligned} \sum _{k=1; p\not \mid k}^{(p^r-1)/2}\frac{1}{k^n}\equiv 0~~({\mathrm{mod}}~p^r). \end{aligned}$$

In addition, we prove the following lemma.

Lemma 2.2

Let $p\ge 5$ be a prime and n be an even integer such that $n\not \equiv 0$ (mod $p-1)$. Then for all integers $r> 0$,

$$\begin{aligned} \sum _{k=1; p\not \mid k}^{p^r-1}\frac{(-1)^k}{k^n}\equiv 0~~({\mathrm{mod}}~p^r). \end{aligned}$$

Proof

If $p\not \mid k$, then $p\not \mid (p^r-k)$ for any $r> 0$. Therefore,

$$\begin{aligned} \sum _{k=1; p\not \mid k}^{p^r-1}\frac{(-1)^k}{k^n} =\sum _{k=1; p\not \mid k}^{\frac{p^r-1}{2}}\left\{ \frac{(-1)^k}{k^n}+\frac{(-1)^{p^r-k}}{(p^r-k)^n}\right\} \equiv \sum _{k=1}^{\frac{p^r-1}{2}}(-1)^k\left\{ \frac{1}{k^n}-\frac{1}{(-k)^n}\right\} ~~(\text {mod}~p^r). \end{aligned}$$

Since n is even, we complete the proof of the lemma. $\square$

Lemma 2.3

For prime p and integers $f_1,f_2,k\ge 0,r\ge 1, A\ge 0,B\ge 0,$ we have

$$\begin{aligned} \left( {\begin{array}{c}f_{2p^{r}}{-1}\\ k\end{array}}\right) ^{A}&\left( {\begin{array}{c}f_{1p^{r}}+k\\ k\end{array}}\right) ^{B} \equiv (-1)^{\left( k+\left[ \frac{k}{p}\right] \right) A}\left( {\begin{array}{c}f_{2p^{r-1}}-1\\ {\left[ \frac{k}{p}\right] }\end{array}}\right) ^{A}\left( {\begin{array}{c}f_{1p^{r-1}}+\left[ \frac{k}{p}\right] \\ {[\frac{k}{p}]}\end{array}}\right) ^B~~({\mathrm{mod}}~p^r). \end{aligned}$$

Proof

The proof of the lemma follows immediately from [24, Lemma 2.5]. $\square$

Proof of Theorem 1.1

Let $\lambda =\pm 1$. Splitting into two sums, we obtain

$$\begin{aligned} A(f_1p^r,f_2p^r,m,l,\lambda )&=\sum _{j=0}^{f_2p^r}\left( {\begin{array}{c}f_{1p^r}+j\\ j\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2p^{r}}\\ j\end{array}}\right) ^{l} \lambda ^j\nonumber \\&=\sum _{j=0;p\not \mid j}^{f_2p^r}\left( {\begin{array}{c}f_{1p^r}+j\\ j\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2p^{r}}\\ j\end{array}}\right) ^{l}\lambda ^j+\sum _{j=0;p\mid j}^{f_2p^r}\left( {\begin{array}{c}f_{1p^{r}}+j\\ j\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2p^{r}}\\ j\end{array}}\right) ^{l}\lambda ^j\nonumber \\&\,{:=}\,S_1+S_2. \end{aligned}$$

(2.1)

If $p\mid j$, then $j=kp^s$ for some $k,s\ge 0$ and $p\not \mid k$. Therefore,

$$\begin{aligned} S_2=\sum _{k=0}^{f_2}\sum _{s=1}^{r}\left( {\begin{array}{c}f_{1p^{r}}+kp^{s}\\ kp^s\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2p^r}\\ kp^s\end{array}}\right) ^{l}\lambda ^k. \end{aligned}$$

Clearly $s\le r$, and hence using Jacobsthal’s congruences [24, Lemma 2.1]

$$\begin{aligned} \left( {\begin{array}{c}f_{2p^{r}}\\ kp^s\end{array}}\right) \equiv \left( {\begin{array}{c}f_{2p^{r-1}}\\ kp^{s-1}\end{array}}\right) (\mathrm{mod}p^{r+2s}) \end{aligned}$$

and

$$\begin{aligned} \left( {\begin{array}{c}f_{1p^{r}}+kp^{s}\\ kp^s\end{array}}\right) \equiv \left( {\begin{array}{c}f_{1p^{r-1}}+kp^{s-1}\\ kp^{s-1}\end{array}}\right) (\mathrm{mod}p^{r+2s}) \end{aligned}$$

we deduce that

$$\begin{aligned} S_2\equiv & {} \sum _{k=0}^{f_2}\sum _{s=1}^{r}\left( {\begin{array}{c}f_{1p^{r-1}}+kp^{s-1}\\ kp^{s-1}\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2p^{r-1}}\\ kp^{s-1}\end{array}}\right) ^{l}\lambda ^{kp^{r+2s-1}} ~~(\mathrm{mod} ~p^{r+2s}) \nonumber \\\equiv & {} \sum _{k=0}^{f_2}\sum _{s=1}^{r}\left( {\begin{array}{c}f_{1p^{r-1}}+kp^{s-1}\\ kp^{s-1}\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2p^{r-1}}\\ kp^{s-1}\end{array}}\right) ^{l} \lambda ^k ~~(\mathrm{mod} ~p^{r+2s}). \end{aligned}$$

(2.2)

For $p\not \mid k$, we have

$$\begin{aligned} \left( {\begin{array}{c}f_{2p^{r}}\\ kp^s\end{array}}\right) ^{l}\equiv \left( {\begin{array}{c}f_{2p^{r-1}}\\ kp^{s-1}\end{array}}\right) ^{l}\equiv 0~~(\mathrm{mod}~p^{l(r-s)}). \end{aligned}$$

In particular,

$$\begin{aligned}&\sum _{k=0}^{f_2}\sum _{s=1}^{r}\left( {\begin{array}{c}f_{1p^{r-1}}+kp^{s-1}\\ kp^{s-1}\end{array}}\right) ^{m} \left( {\begin{array}{c}f_{2p^{r-1}}\\ kp^{s-1}\end{array}}\right) ^{l}\lambda ^k \nonumber \\&\quad \equiv \sum _{k=0}^{f_2}\sum _{s=1}^{r}\left( {\begin{array}{c}f_{1p^r}+kp^{s}\\ kp^{s}\end{array}}\right) ^{m}\left( {\begin{array}{c}f_{2p^{r}}\\ kp^{s}\end{array}}\right) ^{l}\lambda ^k\equiv 0~~(\mathrm{mod} ~p^{2(r-s)}). \end{aligned}$$

(2.3)

Since $r+2s+2(r-s)=3r$, (2.2) and (2.3) give

$$\begin{aligned} S_2=\sum _{j=0}^{f_2p^{r-1}}\left( {\begin{array}{c}f_{1p^{r-1}}+j\\ j\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^{r-1}}\\ j\end{array}}\right) ^{l}\lambda ^j\equiv A(f_1p^{r-1},f_2p^{r-1},m,l,\lambda )~~(\text {mod}~p^{3r}). \end{aligned}$$

(2.4)

Again if $p\not \mid j$, then

$$\begin{aligned} ord_p\left( {\begin{array}{c}f_{2p^r}\\ j\end{array}}\right) ^{l}=ord_p\left( {\begin{array}{c}f_{2p^r}\\ jp^0\end{array}}\right) ^{l}\ge lr. \end{aligned}$$

In particular,

$$\begin{aligned} S_1\equiv 0~~(\text {mod}~p^{3r}) \end{aligned}$$

for $l\ge 3$, and hence the proof follows from (2.1) because of (2.4) when $l\ge 3$. Thus we are left to prove the case when $l = 2$. We now follow the approach of [24]. For $l=2$, we need to show that

$$\begin{aligned} S_1&\equiv \sum _{j=0;p\not \mid j}^{f_2p^r}\left( {\begin{array}{c}f_{1p^r}+j\\ j\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^r}\\ j\end{array}}\right) ^2\lambda ^j\\&\equiv (f_2p^r)^{2}\sum _{j=0;p\not \mid j}^{f_2p^r}\frac{1}{j^2}\left( {\begin{array}{c}f_{1p^r}+j\\ j\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^r}-1\\ j-1\end{array}}\right) ^2 \lambda ^j\equiv 0 ~~(\text {mod} ~p^{3r}), \end{aligned}$$

which is equivalent to

$$\begin{aligned} \sum _{j=0;p\not \mid j}^{f_2p^r}\frac{1}{j^2}\left( {\begin{array}{c}f_{1p^{r}}+j\\ j\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^r}-1\\ j-1\end{array}}\right) ^2\lambda ^j\equiv 0 ~~(\text {mod} ~p^{r}). \end{aligned}$$

(2.5)

Using Lemma 2.3, we obtain

$$\begin{aligned}&\sum _{j=0;p\not \mid j}^{f_2p^r}\frac{1}{j^2}\left( {\begin{array}{c}f_{1p^{r}}+j\\ j\end{array}}\right) ^m \left( {\begin{array}{c}f_{2p^r}-1\\ j-1\end{array}}\right) ^2\lambda ^j\nonumber \\&\quad \equiv \sum _{j=0;p\not \mid j}^{f_2p^{r-1}}\frac{1}{j^2} \left( \begin{array}{c} {f_{1p^{r-1}}+\left[ \frac{j}{p}\right] }\\ {{\left[ \frac{j}{p}\right] }} \end{array}\right) ^m \left( \begin{array}{c} {f_{2p^{r-1}}-1}\\ {{\left[ \frac{j}{p}\right] }} \end{array}\right) ^2\lambda ^j ~~(\text {mod} ~p^{r}) \end{aligned}$$

(2.6)

because of the fact that $\left[ \frac{k-1}{p}\right] = \left[ \frac{k}{p}\right]$ when $p\not \mid k$. In view of (2.5) and (2.6), it is enough to show that

$$\begin{aligned}&\sum _{j=0;p\not \mid j}^{f_2p^r}\frac{1}{j^2}\left( {\begin{array}{c}f_{1p^{r}}+j\\ j\end{array}}\right) ^m \left( {\begin{array}{c}f_{2p^{r}}-1\\ j\end{array}}\right) ^2\lambda ^j\nonumber \\&\quad \equiv \sum _{j=0;p\not \mid j}^{f_2p^{r-s}}\frac{1}{j^2}\left( \begin{array}{c}{f_{1p^{r-s}}}+\left[ \frac{j}{p^s}\right] \\ {{\left[ \frac{j}{p^s}\right] }} \end{array}\right) ^m \left( \begin{array}{c}{f_{2p^{r-s}}-1}\\ {{[\frac{j}{p^s}]}} \end{array}\right) ^2\lambda ^j~~(\text {mod} ~p^{r}) \end{aligned}$$

(2.7)

for $s=0,1,\ldots , r$. The case $s=0$ is trivial, whereas Lemma 2.3 proves it for $s=1$. Let $\lbrace j{:}p^s\rbrace \,{:=}\,j-p^s[\frac{j}{p^s}]$, then

$$\begin{aligned}&\sum _{j=0;p\not \mid j}^{f_2p^{r-s}}\frac{1}{j^2} \left( \begin{array}{c}{f_{1p^{r-s}}+\left[ \frac{j}{p^s}\right] }\\ {{\left[ \frac{j}{n^s}\right] }} \end{array}\right) ^m \left( \begin{array}{c}{f_{2p^{r-s}}-1}\\ {{[\frac{j}{p^s}]}} \end{array}\right) ^2 \lambda ^j \nonumber \\&\quad =\sum _{n}\left( {\begin{array}{c}f_{1p^{r-s}}+n\\ n\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^{r-s}}-1\\ n\end{array}}\right) ^2\sum _{j=0;p\not \mid j;\left[ \frac{j}{p^s}\right] =n}^{f_2p^{r-s}} \frac{\lambda ^j}{j^2}\end{aligned}$$

(2.8)

$$\begin{aligned}&\quad =\sum _{n}\left( {\begin{array}{c}f_{1p^{r-s}}+n\\ n\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^{r-s}}-1\\ n\end{array}}\right) ^2\sum _{j=0;p\not \mid j;\left[ \frac{j}{p^s}\right] =n;\{j{:}p^s\}<p^s/2}^{f_2p^{r-s}}\frac{\lambda ^j}{j^2}\nonumber \\&\qquad +\sum _{n}\left( {\begin{array}{c}f_{1p^{r-s}}+n\\ n\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^{r-s}}-1\\ n\end{array}}\right) ^2\sum _{j=0;p\not \mid j;\left[ \frac{j}{p^s}\right] =n;\{j{:}p^s\}>p^s/2}^{f_2p^{r-s}}\frac{\lambda ^j}{j^2}. \end{aligned}$$

(2.9)

It is clear from Lemma 2.2 that $p^s$ divides the inner sum of (2.8) for $\lambda =-1$, and Lemma 2.1 implies that $p^s$ divides each inner sum of (2.9) for $\lambda =1$. For $s<r$, we use Lemma 2.3 in the above expression to obtain

$$\begin{aligned} \sum _{j=0;p\not \mid j}^{f_2p^{r-s}}&\frac{1}{j^2}\left( \begin{array}{c}{f_{1p^{r-s}}+\left[ \frac{j}{p^s}\right] }\\ {{\left[ \frac{j}{p^s}\right] }}\end{array}\right) ^m \left( \begin{array}{c}{f_{2p^{r-s}}-1}\\ {{\left[ \frac{j}{p^s}\right] }}\end{array}\right) ^2 \lambda ^j\\ \equiv&\sum _{n}\left( \begin{array}{c}{f_{1p^{r-s-1}}+\left[ \frac{n}{p}\right] }\\ {{\left[ \frac{n}{p}\right] }} \end{array}\right) ^m \left( \begin{array}{c}{f_{2p^{r-s-1}}-1}\\ {{\left[ \frac{n}{p}\right] }}\end{array}\right) ^2 \sum _{j=0;p\not \mid j;\left[ \frac{j}{p^s}\right] =n;\{j{:}p^s\}<p^s/2}^{f_2p^{r-s}}\frac{\lambda ^j}{j^2}\\&+ \sum _{n}\left( \begin{array}{c}{f_{1p^{r-s-1}}+\left[ \frac{n}{p}\right] }\\ {{\left[ \frac{n}{p}\right] }}\end{array}\right) ^m \left( \begin{array}{c}{f_{2p^{r-s-1}}-1}\\ {{\left[ \frac{n}{p}\right] }}\end{array}\right) ^2 \sum _{j=0;p\not \mid j;\left[ \frac{j}{p^s}\right] =n;\{j{:}p^s\}>p^s/2}^{f_2p^{r-s}}\frac{\lambda ^j}{j^2}~~(\text {mod}~p^r)\\ \equiv&\sum _{j=0;p\not \mid j}^{f_2p^{r-s-1}}\frac{1}{j^2}\left( \begin{array}{c}{f_{1p^{r-s-1}}+\left[ \frac{j}{p^{s+1}}\right] }\\ {{\left[ \frac{j}{p^{s+1}}\right] }}\end{array}\right) ^m \left( \begin{array}{c}{f_{2p^{r-s-1}}-1}\\ {{\left[ \frac{j}{p^{s+1}}\right] }}\end{array}\right) ^2 \lambda ^j~~(\text {mod}~p^r) \end{aligned}$$

Induction on s completes the proof of (2.7). Moreover, (2.8), (2.9) and (2.7) together imply that

$$\begin{aligned} \sum _{j=0;p\not \mid j}^{f_2p^r}\frac{1}{j^2}\left( {\begin{array}{c}f_{1p^{r}}+j\\ j\end{array}}\right) ^m\left( {\begin{array}{c}f_{2p^{r}}-1\\ j\end{array}}\right) ^2\lambda ^j\equiv 0~~(\text {mod}~p^r), \end{aligned}$$

which completes the proof of the theorem. $\square$

3 Proof of Theorem 1.2 and 1.3

Proof of Theorem 1.2

Let $\omega$ denote the Teichmüller character defined as $\omega (x)\equiv x$ (mod p) for $x = 0,1,\ldots , p-1$. Using [20, Lemma 1], we obtain

$$\begin{aligned} A\left( \frac{a(p-1)}{r},\frac{b(p-1)}{s},m,l,\lambda \right)&=\sum _{j=0}^{\frac{b(p-1)}{s}}\left( \begin{array}{c}{\frac{a(p-1)}{r}+j}\\ {j}\end{array}\right) ^m \left( \begin{array}{c}{\frac{b(p-1)}{s}}\\ {j} \end{array}\right) ^{l}\lambda ^j\\&\equiv \left( \frac{p}{p-1}\right) ^w\sum _{j=0}^{\frac{b(p-1)}{s}}\left( \begin{array}{c}{\omega ^{\frac{a(p-1)}{r}}\omega ^j}\\ {\omega ^j}\end{array}\right) ^m \left( \begin{array}{c}{\omega ^{\frac{b(p-1)}{s}}}\\ {\omega ^j} \end{array}\right) ^{l} \omega ^j(\lambda )~~(\text {mod}~p)\\&\equiv \left( \frac{p}{p-1}\right) ^w\sum _\chi \left( {\begin{array}{c}\psi ^{a}\chi \\ \chi \end{array}}\right) ^m\left( {\begin{array}{c}\xi ^{b}\\ \chi \end{array}}\right) ^{l}\chi (\lambda )~~(\text {mod}~p), \end{aligned}$$

where $\psi$ and $\xi$ are characters of order r and s, respectively. Using (1.2), we have

$$\begin{aligned} A\left( \frac{a(p-1)}{r},\frac{b(p-1)}{s},m,l,\lambda \right)&\equiv \left( \frac{p}{p-1}\right) ^w\sum _\chi \left( {\begin{array}{c}\psi ^{a}\chi \\ \chi \end{array}}\right) ^m\left( {\begin{array}{c}\bar{\xi }^{b}\chi \\ \chi \end{array}}\right) ^{l}\chi (-1)^{l}\chi (\lambda )~~(\text {mod}~p)\\&\equiv \left( \frac{p}{p-1}\right) ^w\sum _\chi \left( {\begin{array}{c}\psi ^{a}\chi \\ \chi \end{array}}\right) ^m\left( {\begin{array}{c}\bar{\xi }^{b}\chi \\ \chi \end{array}}\right) ^{l}\chi ((-1)^{l}\lambda )~~(\text {mod}~p). \end{aligned}$$

Thus the fact $\frac{1}{p-1}\equiv -1$ (mod p) completes the proof of the theorem. $\square$

Proof of Theorem 1.3

Putting $S=\phi$ in each expression of [4, Theorem 1.8], we obtain

$$\begin{aligned}&{_{2}}F_1\left( \begin{array}{cccc} \chi _4, &{} \chi _4^3\\ &{} \epsilon \end{array}\mid \frac{4}{3} \right) =\left\{ \begin{array}{ll} 0, &{} \quad \hbox {if}\,p\equiv 2\, (\mathrm{mod} \,3); \\ \displaystyle \phi (6)\chi _4(-1) \left[ \displaystyle {\phi \atopwithdelims ()\chi _3}+{\phi \atopwithdelims ()\chi _3^2}\right] , &{} \quad \hbox {if}\,p\equiv 1\, (\mathrm{mod} \,3) \end{array} \right. \\&{_{2}}F_1\left( \begin{array}{cccc} \chi _4, &{} \chi _4^3\\ &{} \epsilon \end{array}\mid -\frac{1}{3} \right) =\left\{ \begin{array}{ll} 0, &{} \quad \hbox {if}\,p\equiv 2\, (\mathrm{mod} \,3); \\ \displaystyle \phi (6)\displaystyle \left[ {\phi \atopwithdelims ()\chi _3}+{\phi \atopwithdelims ()\chi _3^2}\right] , &{} \quad \hbox {if}\,p\equiv 1 (\mathrm{mod} \,3) \end{array} \right. \\&{_{2}}F_1\left( \begin{array}{cccc} \chi _4, &{} \chi _4^3\\ &{} \epsilon \end{array}\mid 4 \right) =\left\{ \begin{array}{ll} 0, &{} \quad \hbox {if}\,p\equiv 2\, (\mathrm{mod} \,3); \\ \displaystyle \chi _4(12)\displaystyle \left[ {\phi \atopwithdelims ()\chi _3}+{\phi \atopwithdelims ()\chi _3^2}\right] ,&{} \quad \hbox {if}\,p\equiv 1 \,(\mathrm{mod}\, 3) \end{array} \right. \\&{_{2}}F_1\left( \begin{array}{cccc} \chi _4, &{} \chi _4\\ &{} \epsilon \end{array}\mid \frac{1}{4} \right) =\left\{ \begin{array}{ll} 0, &{} \quad \hbox {if}\,p\equiv 2\, (\mathrm{mod} \,3); \\ \displaystyle \chi _4(3) \displaystyle \left[ {\phi \atopwithdelims ()\chi _3}+{\phi \atopwithdelims ()\chi _3^2}\right] , &{} \quad \hbox {if}\,p\equiv 1 (\mathrm{mod}\, 3), \end{array} \right. \end{aligned}$$

respectively. Together with Theorem 1.2 and [8, Section 3.1], the first two expressions yield (i) and the last two complete the proof of (ii). $\square$

In view of [6, Theorem 1.4], we can also deduce the following congruences similar to Theorem 1.3.

Corollary 3.1

Let $p=a_4^2+b_4^2\equiv 1$ (mod 4), where $a_4\equiv -\phi (2)$ (mod 4). If $d=\frac{p-1}{4}$, then

Similarly, [7, Theorem 1.7 and Corollary 3.4] provides the following corollary.

Corollary 3.2

If $p\equiv 1$ (mod 3) and $e=\frac{p-1}{3}$, then

4 Preliminaries on p-adic gamma function

In this section we recall some notations and preliminary results required to prove Theorem 1.4. We begin with properties of p-adic Gamma function $\Gamma _p$ (for details, see [19]). For $n\in \mathbb {N}$, the p-adic Gamma function is defined as

$$\begin{aligned} \Gamma _p(n)=(-1)^n\prod _{0<j<n, p\not \mid j}j. \end{aligned}$$

Extend $\Gamma _p$ to all $x\in \mathbb {Z}_p$ by setting $\Gamma _p(0)=1$ and

$$\begin{aligned} \Gamma _p(x)=\lim _{n\rightarrow x}\Gamma (n), \end{aligned}$$

where n runs through any sequence of positive integers which p-adically approaches x. The following lemma provides some main properties of Gamma function, which are easy consequences of its definition.

Lemma 4.1

[1, 19] Let $p\ge 5$ be a prime. If $x,y,z \in \mathbb {Z}_p$ with $|z|\le |p|$, where $|\cdot |$ is the p-adic norm. Then

• (a) $\Gamma _p(x+1)=\left\{ \begin{array}{ll} -x\Gamma _p(x), &{} \quad \hbox {if}\,|x|=1; \\ -\Gamma _p(x), &{} \quad \hbox {if}\,|x|<1. \\ \end{array} \right.$

• (b) $\Gamma _p(x)\Gamma _p(1-x)=(-1)^{\text {Re}(x)}$, where Re(x) denotes the representative of x (mod p) in the set $\{1,2,\ldots ,p\}$.

• (c) if $x\equiv y$ (mod $p^n)$, then $\Gamma _p(x)\equiv \Gamma (y)$ (mod $p^n)$; $n\in \mathbb {N}$.

• (d) $n!=(-1)^{n+1}\Gamma _p(n+1)$; $0\le n\le (p-1)$.

• (e) $\Gamma _p'(x+z)\equiv \Gamma _p'(x)$ (mod p).

• (f) $\Gamma _p(x+z)\equiv \Gamma _p(x)+z\Gamma _p'(x)$ (mod $p^2)$.

We consider the logarithmic derivative given by $G(x)\,{:=}\,\frac{\Gamma _p'(x)}{\Gamma _p(x)}$. It is easy to see that if $x\in \mathbb {Z}_p$, then $G(x)\in \mathbb {Z}_p$. Moreover if $x\in \mathbb {Z}_p$ and $|x|=1$, then the logarithmic derivative G(x) satisfy the following nice relation

$$\begin{aligned} G(x+1)-G(x)=\frac{1}{x}. \end{aligned}$$

(4.1)

Finally, we recall some background on Gauss sums. Let $\pi \in \mathbb {C}_p$ be a fixed root of $x^{p-1}+p=0$, and we let $\zeta _p$ be the unique pth root of unity in $\mathbb {C}_p$ such that $\zeta _p\equiv 1+\pi$ (mod $\pi ^2)$. For a multiplicative character $\chi$, we define the Gauss sum to be

$$\begin{aligned} g(\chi )=\sum _{x=0}^{p-1}\chi (x)\zeta _p^x. \end{aligned}$$

The following lemma provide some well-known properties of Gauss sums:

Lemma 4.2

[8] For characters $\chi ,\chi _1,\chi _2$ on $\mathbb {F}_p$, we have

• (a) $g(\chi )g(\overline{\chi }) = \chi (-1)p$.

• (b) $J(\chi _1, \chi _2) = \left\{ \begin{array}{ll} \frac{g(\chi _1)g(\chi _2)}{g(\chi _1\chi _2)}, &{} \quad \hbox {if}\, \chi _1\chi _2\ne \varepsilon ; \\ -\chi _1(-1), &{} \quad \hbox {if}\,\chi _1\chi _2=\varepsilon \quad \mathrm{and} \quad \chi _1\ne \varepsilon ,\chi _2\ne \varepsilon . \end{array} \right.$

Let $\omega$ denote the Teichmüller character, then $\omega$ can be defined uniquely by the property that $\omega (x)\equiv x$ (mod p) for $x = 0,\ldots , p-1$. In this context, the Gross-Koblitz formula [15] states that

$$\begin{aligned} g(\overline{\omega }^j)=-\pi ^j\Gamma _p\left( \frac{j}{p-1}\right) , \quad 0\le j\le p-2. \end{aligned}$$

(4.2)

5 Proof of Theorem 1.4

The main aim of this section is to prove Theorem 1.4. Following the approach of [1], we break the proof into a number of lemmas, and then combine them to prove the result.

Lemma 5.1

Let $p>3$ be a prime such that $p\equiv 1$ (mod r) and ord$_p(\lambda )>0$. If $T\in \widehat{\mathbb {F}_q^\times }$ is a generator of the character group, then

$$\begin{aligned}&-p^{w-1}{_w}F_{w-1}\left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{l}\lambda \right) \\&\quad \equiv \frac{1}{\Gamma _p\left( \frac{1}{r}\right) ^w}\sum _{j=0}^{\frac{p-1}{r}}\frac{\lambda ^{pj}\Gamma _p\left( \frac{1}{r}+j\right) ^w}{(-1)^{jl}\Gamma _p(1+j)^w} \left\{ 1+p\left\{ 1+wj\left\{ G\left( \frac{1}{r}+j\right) -G(1+j)\right\} \right\} \right\} ~~(\mathrm{{mod}}~p^2). \end{aligned}$$

Proof

By definition of Gaussian hypergeometric series, we have

$$\begin{aligned}&{_w}F_{w-1}\left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \quad \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{l}\lambda \right) \\&\quad =\frac{p}{p-1}\sum _{\chi }\left( {\begin{array}{c}{T^{\frac{p-1}{r}}}\chi \\ \chi \end{array}}\right) ^w\chi (-1)^{l}\chi (\lambda )\\&\quad =\frac{1}{p^{w-1}(p-1)}\sum _{\chi }J\left( T^{\frac{p-1}{r}}{\chi },{\bar{\chi }}\right) ^w\chi (-1)^{l+w}\chi (\lambda ). \end{aligned}$$

By Lemma 4.2, we obtain

$$\begin{aligned}&p^{w-1} {_w}F_{w-1}\left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{l}\lambda \right) \\&\quad =\frac{1}{p-1}\sum _{j=0}^{p-2}J({\bar{\omega }}^{\frac{p-1}{r}}{{\bar{\omega }}^{-j}},{\bar{\omega }}^j)^w{\bar{\omega }}^j(-1)^{l+w}\omega ^j(\lambda )\\&\quad =\frac{1}{p-1}\sum _{j=0}^{p-2}\frac{g({\bar{\omega }}^{\frac{p-1}{r}-j})^wg({\bar{\omega }}^j)^w }{g({\bar{\omega }}^{\frac{p-1}{r}})^w}{\bar{\omega }}^{j(l+w)}(-1)\omega ^j(\lambda )\\&\quad =\frac{1}{p-1}\sum _{j=0}^{\frac{p-1}{r}}\frac{g({\bar{\omega }}^{\frac{p-1}{r}-j})^wg({\bar{\omega }}^j)^w }{g({\bar{\omega }}^{\frac{p-1}{r}})^w}{\bar{\omega }}^{j(l+w)}(-1)\omega ^j(\lambda )\\&\qquad +\frac{1}{p-1}\sum _{j=\frac{p-1}{r}+1}^{p-2}\frac{g({\bar{\omega }}^{\frac{r+1}{r}(p-1)-j})^wg({\bar{\omega }}^j)^w }{g({\bar{\omega }}^{\frac{p-1}{r}})^w}{\bar{\omega }}^{j(l+w)}(-1)\omega ^j(\lambda ), \end{aligned}$$

where $\omega$ is the Teichmüller character. Using the Gross-Koblitz formula (4.2) and the fact that $\pi ^{w(p-1)}=(-p)^w$, we deduce

$$\begin{aligned}&p^{w-1} {_w}F_{w-1}\left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{l}\lambda \right) \\&\quad =\frac{(-1)^w}{p-1}\sum _{j=0}^{\frac{p-1}{r}}\frac{\Gamma _p(\frac{1}{r}-\frac{j}{p-1})^w\Gamma _p(\frac{j}{p-1})^w}{\Gamma _p(\frac{1}{r})^w}{\bar{\omega }}^{j(l+w)}(-1)\omega ^j(\lambda )\\&\qquad +\frac{(-1)^w}{p-1}\sum _{j=\frac{p-1}{r}+1}^{p-2}\frac{\pi ^{w(p-1)}\Gamma _p(\frac{r+1}{r}-\frac{j}{p-1})^w\Gamma _p(\frac{j}{p-1})^w}{\Gamma _p(\frac{1}{r})^w}{\bar{\omega }}^{j(l+w)}(-1)\omega ^j(\lambda )\\&\quad \equiv \frac{1}{p-1}\sum _{j=0}^{\frac{p-1}{r}}\frac{(-1)^w\Gamma _p(\frac{1}{r}+\frac{j}{1-p})^w\Gamma _p(\frac{-j}{1-p})^w}{\Gamma _p(\frac{1}{r})^w}{\bar{\omega }}^{j(l+w)}(-1)\omega ^j(\lambda )~~(\text {mod}~p^2). \end{aligned}$$

It is known that $\frac{j}{1-p}\equiv j+jp$ (mod $p^2)$ and $\omega (\lambda )=\lambda ^p$ (mod $p^2)$. Using these together with Lemma 4.1 (b), we have

$$\begin{aligned}&p^{w-1} {_w}F_{w-1}\left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{l}\lambda \right) \\&\quad \equiv \frac{1}{p-1}\sum _{j=0}^{\frac{p-1}{r}}\frac{(-1)^w\Gamma _p(\frac{1}{r}+j+jp)^w\Gamma _p(-j-jp)^w}{\Gamma _p(\frac{1}{r})^w}{\bar{\omega }}^{j(l+w)}(-1)\omega ^j(\lambda )~~(\text {mod}~p^2)\\&\quad \equiv \frac{1}{p-1}\sum _{j=0}^{\frac{p-1}{r}}\frac{\Gamma _p(\frac{1}{r}+j+jp)^w(-1)^{jl}}{\Gamma _p(\frac{1}{r})^w\Gamma _p(1+j+jp)^w} \lambda ^{jp}~~(\text {mod}~ p^2)\\&\quad \equiv -\frac{(1+p)}{\Gamma _p(\frac{1}{r})^w}\sum _{j=0}^{\frac{p-1}{r}}\frac{\Gamma _p(\frac{1}{r}+j+jp)^w}{\Gamma _p(1+j+jp)^w}(-1)^{jl}\lambda ^{jp}~~(\text {mod}~ p^2). \end{aligned}$$

Finally, Lemma 4.1 (f) yields

$$\begin{aligned}&-p^{w-1}{_w}F_{w-1}\left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{l}\lambda \right) \\&\quad \equiv \frac{(1+p)}{\Gamma _p(\frac{1}{r})^w}\sum _{j=0}^{\frac{p-1}{r}} \frac{\Gamma _p(\frac{1}{r}+j)^w\left\{ 1+wjpG(\frac{1}{r}+j)\right\} }{\Gamma _p(1+j)^w\left\{ 1+wjpG(1+j)\right\} }(-1)^{jl}\lambda ^{jp}~~(\text {mod}~p^2). \end{aligned}$$

Multiplying numerator and denominator by $\lbrace 1-wjpG(1+j)\rbrace$, and then simplifying we complete the proof of the lemma. $\square$

Lemma 5.2

Let $p>3$ be a prime for which ord$_p(\lambda )>0$, then

$$\begin{aligned} A&\left( \frac{(r-1)(p-1)}{r},\frac{p-1}{r},m,m(r-1),\lambda ^p\right) \\&\equiv \frac{(-1)^{mr}}{\Gamma _p(\frac{1}{r})^{mr}}\sum _{j=0}^{\frac{p-1}{r}} \frac{\Gamma _p(\frac{1}{r}+j)^{mr}}{\Gamma _p(1+j)^{mr}}(-1)^{jm(r-1)}\lambda ^{pj}~~({\mathrm{mod}}~~p^2). \end{aligned}$$

Proof

Using Lemma 4.1 (d) and (b), we have

$$\begin{aligned} A&\left( \frac{(r-1)(p-1)}{r},\frac{p-1}{r},m,m(r-1),\lambda ^p\right) \\&=\sum _{j=0}^{\frac{p-1}{r}}\left( {\begin{array}{c}\frac{(r-1)(p-1)}{r}+j\\ j\end{array}}\right) ^m \left( {\begin{array}{c}\frac{p-1}{r}\\ j\end{array}}\right) ^{m(r-1)}\lambda ^{pj}\\&=\sum _{j=0}^{\frac{p-1}{r}}\frac{{(\frac{(r-1)(p-1)}{r}+j)!}^m}{{j!}^m{\frac{(r-1)(p-1)}{r}!}^m}\times \frac{{(\frac{p-1}{r})!}^{m(r-1)}}{{j!}^{m(r-1)} {(\frac{p-1}{r}-j)!}^{m(r-1)}}\lambda ^{pj}\\&=\frac{\Gamma _p(\frac{p-1}{r}+1)^{m(r-1)}}{(-1)^{mr}\Gamma _p(\frac{(r-1)(p-1)}{r}+1)^m}\sum _{j=0}^{\frac{p-1}{r}}\frac{\Gamma _p(\frac{(r-1)(p-1)}{r}+j+1)^m}{\Gamma _p(1+j)^{mr} \Gamma _p(\frac{p-1}{r}-j+1)^{m(r-1)}}\lambda ^{pj}\\&=\frac{(-1)^{mr}}{\Gamma _p(\frac{r-1}{r}p+\frac{1}{r})^m\Gamma _p(\frac{1}{r}-\frac{p}{r})^{m(r-1)}} \sum _{j=0}^{\frac{p-1}{r}}\frac{\Gamma _p(\frac{r-1}{r}p+j+\frac{1}{r})^m\Gamma _p(\frac{1}{r}+j-\frac{p}{r})^{m(r-1)}}{\Gamma _p(1+j)^{mr}(-1)^{mj(r-1)}}\lambda ^{pj}. \end{aligned}$$

Finally, using Lemma 4.1 (f) and then simplifying we complete the proof of the lemma. $\square$

Proof of Theorem 1.4

If $l=m(r-1)$, then Lemma 5.1 yields

$$\begin{aligned}&p^{mr-1} {_{mr}}F_{mr-1}\left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \cdots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \cdots , &{} \epsilon \end{array}\mid (-1)^{m(r-1)}\lambda \right) \\&\quad \equiv -\frac{1}{\Gamma _p(\frac{1}{r})^{mr}}\sum _{j=0}^{\frac{p-1}{r}}\frac{\Gamma _p(\frac{1}{r}+j)^{mr}}{\Gamma _p(1+j)^{mr}}(-1)^{jm(r-1)}\lambda ^{pj}-\frac{p}{\Gamma _p(\frac{1}{r})^{mr}}\sum _{j=0}^{\frac{p-1}{r}}\frac{\Gamma _p(\frac{1}{r}+j)^{mr}}{\Gamma _p(1+j)^{mr}}\\&\qquad \times \left\{ 1+wj\left\{ G(\frac{1}{r}+j)-G(1+j)\right\} \right\} (-1)^{jm(r-1)}\lambda ^{pj}~~(\mathrm{{mod}}~p^2). \end{aligned}$$

It is easy to deduce that

$$\begin{aligned} \left( {\begin{array}{c}\frac{r-1}{r}(p-1)+j\\ j\end{array}}\right) ^m\left( {\begin{array}{c}\frac{p-1}{r}\\ j\end{array}}\right) ^{m(r-1)}\equiv \frac{(-1)^{mr}}{\Gamma _p(\frac{1}{r})^{mr}}\frac{\Gamma _p(\frac{1}{r}+j)^{mr}}{\Gamma _p(1+j)^{mr}}(-1)^{jm(r-1)}~~(\text {mod}~p). \end{aligned}$$

Again, repeated application of (4.1) yields

$$\begin{aligned} G\left( \frac{1}{r}+j\right) -G(1+j)\equiv G\left( \frac{p+1}{r}+j\right) -G(1+j)\equiv H_{\frac{p-r+1}{r}+j}-H_j~ (\text {mod}~ p). \end{aligned}$$

Thus Lemma 5.2 together with the fact $\lambda ^p\equiv \lambda$ (mod p), we have

$$\begin{aligned}&(-p)^{mr-1}{_{mr}}F_{mr-1} \left( \begin{array}{cccc} T^{\frac{p-1}{r}}, &{} T^{\frac{p-1}{r}}, &{} \ldots , &{} T^{\frac{p-1}{r}}\\ &{} \epsilon , &{} \ldots , &{} \epsilon \end{array}\mid (-1)^{m(r-1)}\lambda \right) \\&\quad \equiv \, A\left( \frac{r-1}{r}(p-1),\frac{p-1}{r},m,m(r-1),\lambda ^p\right) \\&\qquad +pB\left( \frac{r-1}{r}(p-1),\frac{p-1}{r},m,m(r-1),\lambda \right) ~~(\text {mod}~p^2) \end{aligned}$$

completing the proof of the theorem. $\square$

6 More supercongruences

In [2], Ahlgren proved (1.4) and recorded certain new supercongruences. We also find some similar supercongruences. In the following theorem, we denote by $D_2(\lambda )$ the discriminant of the polynomial $f(x)=x^{4}-4x^2-4\lambda$; $\lambda \ne 0,-1$.

Theorem 6.1

Let $p>3$ be a prime and $t\in \{1,-1\}$.

• (i) If $p=a_4^2+b_4^2\equiv 1$ (mod 8) and $a_4\equiv -\phi (2)\equiv =-1$ (mod 4) such that $p\not \mid D_2(-1/9)$, then

  $$\begin{aligned}&A\left( \frac{p-1}{2},\frac{p-1}{2},1,1,-\left( \frac{4\sqrt{2}}{2\sqrt{2}+3t}\right) ^p\right) +pB&\left( \frac{p-1}{2},\frac{p-1}{2},1,1,-\frac{4\sqrt{2}}{2\sqrt{2}+3t}\right) \\&\equiv -2a_4\phi (3+2\sqrt{2}t)~~({\mathrm{mod}}~p^2). \end{aligned}$$

• (ii) If $p\not \mid D_2(1/3)$, then

  $$\begin{aligned}&A\left( \frac{p-1}{2},\frac{p-1}{2},1,1,-\left( \frac{4}{2+\sqrt{3}t}\right) ^p\right) +pB\left( \frac{p-1}{2},\frac{p-1}{2},1,1,-\frac{4}{2+\sqrt{3}t}\right) \\&\quad \equiv \left\{ \begin{array}{ll} 0~(\mathrm {mod}~p^2), &{} \quad \hbox {if}\,p\equiv 11 (\mathrm{{mod}} 12); \\ -2a_3\phi (3+2\sqrt{3}t) ~(\mathrm {mod}~p^2), &{} \quad \text {if } p=a_3^2+3b_3^2\equiv 1~ (\mathrm {mod}~ 12)\\ &{}\text {and}~ a_3\equiv -1~ (\mathrm{mod}~ 3). \end{array} \right. \end{aligned}$$

Proof

In [18, Theorem 1.1], ${_{2}}F_1(\lambda )$ was evaluated for $\lambda \in \left\{ \frac{4\sqrt{2}}{2\sqrt{2}+3},\frac{4\sqrt{2}}{2\sqrt{2}-3},\frac{4}{2+\sqrt{3}}, \frac{4}{2-\sqrt{3}}\right\}$. Combining these evaluations with Theorem 1.4, we obtain the supercongruences of Theorem 6.1. $\square$

Acknowledgements

We thank Ken Ono for going through the initial draft of the paper and many helpful suggestions during the preparation of the paper. We are grateful to the referee for his/her helpful comments.

Competing interests

The authors declare that they have no competing interests.