1. Introduction

In 1880, Appell defined four kinds of double series $F_{1}$ , $F_{2}$ , $F_{3}$ , $F_{4}$ in two variables (see [Reference Slater12, pages 210–211]) by generalising the Gauss hypergeometric series. These four series, called Appell series, are well known in the field of double hypergeometric series.

Based on the definition of the truncated hypergeometric series, Liu [Reference Liu9] introduced the truncated Appell series, defined by

$$ \begin{align*} F_{1} [a;b,b{'} ;c;x,y]_{n}&=\sum_{i=0}^{n} \sum_{j=0}^{n}\frac{(a)_{i+j}(b)_{i} (b{'})_{j}}{(c)_{i+j}} \cdot \frac{x^{i}y^{j}}{i!j!}; \\ F_{2} [a;b,b{'} ;c,c{'};x,y]_{n}&=\sum_{i=0}^{n} \sum_{j=0}^{n}\frac{(a)_{i+j}(b)_{i} (b{'})_{j}}{(c)_{i}(c{'})_{j}} \cdot \frac{x^{i}y^{j}}{i!j!}; \\ F_{3} [a,a{'};b,b{'} ;c;x,y]_{n}&=\sum_{i=0}^{n} \sum_{j=0}^{n}\frac{(a)_{i}(a{'})_{j}(b)_{i} (b{'})_{j}}{(c)_{i+j}} \cdot \frac{x^{i}y^{j}}{i!j!}; \\ F_{4} [a;b;c,c{'};x,y]_{n}&=\sum_{i=0}^{n} \sum_{j=0}^{n}\frac{(a)_{i+j}(b)_{i+j}}{(c)_{i}(c{'})_{j}} \cdot \frac{x^{i}y^{j}}{i!j!}, \end{align*} $$

where $(x)_n$ is the shifted factorial $(x)_{n} =x(x+1)\cdots (x+n-1)$ with $n \in \mathbb {Z}^{+}$ and $(x)_{0}=1$ .

In [Reference Liu9], Liu confirmed two congruences for the truncated Appell series $F_{1}$ and $F_{2}$ by using combinatorial identities: for any prime $p \ge $ 5, modulo $p^{2}$ ,

$$ \begin{align*} F_1[\tfrac12;\tfrac12,\tfrac12;1;1,1]_{(p-1)/2} &\equiv 1 \\ F_2[\tfrac12;\tfrac12,\tfrac12;1,1;1,1]_{(p-1)/2} &\equiv\begin{cases} -\Gamma_p(\frac{1}{4})^{4} & \mbox{if } p \equiv 1 \pmod{4}, \\ 0 & \mbox{if } p \equiv 3 \pmod{4}. \end{cases} \end{align*} $$

Later, Lin and Liu [Reference Lin and Liu8] studied congruence properties of the truncated Appell series $F_{3}$ and $F_{4}$ : for any prime $p \ge $ 5, modulo $p^{2}$ ,

(1.1) $$ \begin{align} F_3[\tfrac12,\tfrac12;\tfrac12,\tfrac12;1;1,1]_{(p-1)/2} &\equiv (-1)^{{p-1}/{2}},\\F_4[\tfrac12;\tfrac12;1,1;1,1]_{(p-1)/2} &\equiv \begin{cases} (-1)^{{p+1}/{2} }\Gamma_p(\frac{1}{6})^{2}\Gamma_p(\frac{1}{3}) ^{2} & \mbox{if } p \equiv 1 \pmod{3}, \\ 0 & \mbox{if } p \equiv 2 \pmod{3}.\notag \end{cases} \end{align} $$

Here $\Gamma _p$ is the p-adic Gamma function for p an odd prime, given by

$$ \begin{align*} \Gamma_p(\alpha)=\lim_{n\rightarrow \alpha}(-1)^n\prod_{\substack{0<j<n\\ p\nmid j}}j \end{align*} $$

for $\alpha \in \mathbb Z_p$ , and $\mathbb Z_p$ denotes the ring of all p-adic integers.

Recently, Wang and Yu [Reference Wang and Yu14] gave a generalisation of (1.1) with one free parameter d by establishing a q-supercongruence: for n a positive odd integer and d an integer with $n\ge \max \{2d+1, 1-2d \}$ , modulo $\Phi _n(q)^{4}$ ,

$$ \begin{align*} &\sum_{i=0}^{{(n-1)}/{2}-d}\sum_{j=0}^{{(n-1)}/{2}+d}\frac{(q^{2d+1};q^2)_{i}^2(q^{1-2d};q^2)_{j}^2}{(q^2;q^2)_i(q^2;q^2)_j(q^2;q^2)_{i+j}}q^{2ij-4di+4dj} \\ & \equiv \left \{\begin{aligned} & (-1)^{{(n-1)}/{2}}q^{{(1-n^2)}/{4}}, & d = 0, \\ & (1-q^n)^2q^{|d|(2+3|d|-n)-n+{(1-n^2)}/{4}}\sum_{k=1}^{2|d|}(-1)^{k-|d|+{(n-1)}/{2}}q^{k^2-k}H_k(-2|d|-1)\\ & \quad\quad\times\frac{(q^{n+2|d|-2k+1};q^2)_k(q^{4|d|-2k+2};q^2)_{(n-2|d|-1)/2}}{(q^2;q^2)_k(q^2;q^2)_{(n-2|d|-1)/2}}, & d\neq 0,\end{aligned}\right.\end{align*} $$

where $H_k(x)=\sum _{t=1}^{k}{q^{2t+x}}/{(1-q^{2t+x})^2}$ , $k \in \mathbb {Z}^{+}$ . The q-shifted factorial is defined by $(a;q)_{0}=1$ and $(a;q)_{n}=(1-a)(1-aq)\cdots (1-aq^{n-1})$ with $n \in \mathbb {Z}^{+}$ ; the q-integer is $[n]=[n]_{q}=(q^n-1)/(q-1)$ and $\Phi _n(q)$ denotes the nth cyclotomic polynomial in q, which can be factorised as

$$ \begin{align*} \Phi_n(q)=\prod_{\substack{1\leqslant k\leqslant n\\ \gcd(n,k)=1}}(q-\zeta^k) \end{align*} $$

with $\zeta $ a primitive nth root of unity. In addition, the q-binomial coefficient is defined by

$$ \begin{align*} {x\brack k}={x\brack k}_{q} = \begin{cases} & \frac{(q^{1+x-k};q)_k}{(q;q)_k}, & k \geq 0, & \\ & 0, & k < 0. & \end{cases}\end{align*} $$

Inspired by the work mentioned above, and recent progress on congruences and q-congruences (see [Reference Guo2–Reference He and Wang7, Reference Liu and Petrov10, Reference Liu and Wang11, Reference Wang and Xu13–Reference Wang and Yu15]), we continue the study of congruences for the truncated Appell series $F_{1}$ and obtain new results.

Theorem 1.1. Let p be a prime with $p \equiv 1\pmod {4}$ . Then

$$ \begin{align*} F_1[\tfrac12;\tfrac14,\tfrac14;1;1,1]_{(p-1)/2} \equiv 1 \pmod {p^2}. \end{align*} $$

We establish two generalised q-analogues of Theorem 1.1.

Theorem 1.2. Let d and n be positive integers with $n\equiv 1 \pmod {2d}$ . Then, modulo $\Phi _n(q)^{2}$ ,

(1.2) $$ \begin{align} \sum_{i=0}^{{(n-1)}/{d}}\sum_{j=0}^{{(n-1)}/{d}}\frac{(q^{2};q^{2d})_{i+j}(q;q^{2d})_{i}(q;q^{2d})_{j}}{(q^{2d};q^{2d})_{i+j}(q^{2d};q^{2d})_{i}(q^{2d};q^{2d})_{j}}q^{i+2di+2dj} \equiv \begin{bmatrix} -\dfrac{2}{d} \\[2mm] \dfrac{n-1}{d} \end{bmatrix} _{q^{2d}}q^{{(2n^{2}-2)}/{d}}. \end{align} $$

Theorem 1.3. Let d be an even positive integer and n a positive integer with $n \equiv d-1 \pmod {2d}$ . Then, modulo $\Phi _n(q)^{2}$ ,

$$ \begin{align*} &\sum_{i=0}^{{(n-(d-1))}/{d}}\sum_{j=0}^{{(n-(d-1))}/{d}}\frac{(q^{2d-2};q^{2d})_{i+j}(q^{d-1};q^{2d})_{i}(q^{d-1};q^{2d})_{j}}{(q^{2d};q^{2d})_{i+j}(q^{2d};q^{2d})_{i}(q^{2d};q^{2d})_{j}} q^{3di-i+2dj+2d-4} \notag \\ &\qquad \equiv \begin{bmatrix} \dfrac{2}{d} -2 \\[2mm] \dfrac{n-(d-1)}{d} \end{bmatrix} _{q^{2d}}q^{{(2n^{2}-2)}/{d}}. \end{align*} $$

Letting n be a prime p and then taking $q \rightarrow 1$ in Theorems 1.2 and 1.3 gives the following conclusions.

Corollary 1.4. Let p be a prime and d a positive integer with $p \equiv 1 \pmod {2d}$ . Then

$$ \begin{align*} F_1\bigg[\frac1d;\frac{1}{2d},\frac{1}{2d};1;1,1\bigg]_{(p-1)/d} \equiv \begin{pmatrix} -\dfrac{2}{d} \\[2mm] \dfrac{p-1}{d} \end{pmatrix} \pmod {p^2}. \end{align*} $$

Corollary 1.5. Let p be a prime and d an even positive integer with $p \equiv d-1 \pmod {2d}$ . Then

$$ \begin{align*} F_1\bigg[\frac{d-1}{d};\frac{d-1}{2d},\frac{d-1}{2d};1;1,1\bigg]_{(p-(d-1))/d} \equiv \begin{pmatrix} \dfrac{2}{d}-2 \\[2mm] \dfrac{p-(d-1)}{d} \end{pmatrix} \pmod {p^2}. \end{align*} $$

Theorem 1.1 is the special case $d=2$ of Corollaries 1.4 and 1.5. In the following two sections, we give the proofs of Theorems 1.2 and 1.3.

The famous q-Chu–Vandermonde identity [Reference Gasper and Rahman1, (1.5.2)] can be converted to

(1.3) $$ \begin{align}\sum_{k=0}^{n} {x\brack k} {y\brack n-k}q^{(x-k)(n-k)} ={x+y\brack n}, \end{align} $$

and this will be frequently used in our proofs.