11 problems

For any complex numbers $x$ and $q$, define the $q$-shifted factorial by … and, for $n\in\mathbb{Z}^{+}$, … Write $(x_1,x_2,\dots,x_m;q)_n=(x_1;q)_n\cdots(x_m;q)_n$, let…

For integers $d\geq2$ and $n$ satisfying $n\equiv-1\pmod d$ and $n\geq2d-1$, define the rising factorial by $(a)_k=a(a+1)\cdots(a+k-1)$ for $k\geq1$ and $(a)_0=1$. Wang–Li–Tang's i…

Let $n\equiv 1\pmod{4}$ be a positive integer and let $r\geqslant 1$. Write $(a;q)_k$ for the $q$-shifted factorial, $[m]=(1-q^m)/(1-q)$, and let $\Phi_m(q)$ denote the $m$th cyclo…

Let $n\equiv 5\pmod{8}$ be a positive integer, let $e$ be an indeterminate, write $(a;q)_k$ for the $q$-shifted factorial and $[m]=(1-q^m)/(1-q)$, and let $\Phi_n(q)$ denote the…

Let $d\geqslant 2$ be an even integer, let $e$ be an indeterminate, and let $n\equiv d+1\pmod {2d}$ be a positive integer. Write $(a;q)_k$ for the $q$-shifted factorial,…

Let $n equiv 1pmod{4}$ be a positive integer. Define the $q$-integer $[m]=(1-q^m)/(1-q)$, the $q$-shifted factorial $(a;q)k=prod{j=0}^{k-1}(1-aq^j)$, and let $Phin(q)$ denote the…

Guo–Zudilin's conjecture. One has

Let $p,m,r$ satisfy the hypotheses of the source's Theorem 2k2k2k-1. Further q-supercongruence conjecture. … The supplied text contains only the cross-reference to the theorem's hy…

Let $p$ be an odd prime, let $m$ and $|r|$ be positive integers with $p\nmid m$ and $m\nmid r$, and write $\langle a\rangle_p$ for the least nonnegative residue modulo $p$. Exponen…

Let the congruences denoted by (new-RV2)--(new-RV4) be the three earlier q-supercongruences referenced by the source. Strengthening conjecture. Those congruences also hold modulo…

Let $p$ be an odd prime, let $m,r$ be positive integers with $p\nmid m$, and let $s\leqslant p-1$ be nonnegative. Write $\langle a\rangle_p$ for the least nonnegative residue of…