Congruences concerning Jacobi polynomials and Ap\'ery-like formulae

Let $p>5$ be a prime. We prove congruences modulo $p^{3-d}$ for sums of the general form $\sum_{k=0}^{(p-3)/2}\binom{2k}{k}t^k/(2k+1)^{d+1}$ and $\sum_{k=1}^{(p-1)/2}\binom{2k}{k}t^k/k^d$ with $d=0,1$. We also consider the special case $t=(-1)^{d}/16$ of the former sum, where the congruences hold modulo $p^{5-d}$.Comment: to appear in Int. J. Number Theor

Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{\Phi_n(q)},$$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for $a,m\geq 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to appear in Adv. Appl. Mat

On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums

In this work we continue the investigation about the interplay between hypergeometric functions and Fourier-Legendre ($\textrm{FL}$) series expansions. In the section "Hypergeometric series related to $\pi,\pi^2$ and the lemniscate constant", through the FL-expansion of $\left[x(1-x)\right]^\mu$ (with $\mu+1\in\frac{1}{4}\mathbb{N}$) we prove that all the hypergeometric series $$\sum_{n\geq 0}\frac{(-1)^n(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,$$ $$\quad \sum_{n\geq 0}\frac{(4n+1)}{p(n)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^4,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^3,\; \sum_{n\geq 0}\frac{1}{p(n)}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2$$ return rational multiples of $\frac{1}{\pi},\frac{1}{\pi^2}$ or the lemniscate constant, as soon as $p(x)$ is a polynomial fulfilling suitable symmetry constraints. Additionally, by computing the FL-expansions of $\frac{\log x}{\sqrt{x}}$ and related functions, we show that in many cases the hypergeometric $\phantom{}_{p+1} F_{p}(\ldots , z)$ function evaluated at $z=\pm 1$ can be converted into a combination of Euler sums. In particular we perform an explicit evaluation of $$\sum_{n\geq 0}\frac{1}{(2n+1)^2}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2,\quad \sum_{n\geq 0}\frac{1}{(2n+1)^3}\left[\frac{1}{4^n}\binom{2n}{n}\right]^2.$$ In the section "Twisted hypergeometric series" we show that the conversion of some $\phantom{}_{p+1} F_{p}(\ldots,\pm 1)$ values into combinations of Euler sums, driven by FL-expansions, applies equally well to some twisted hypergeometric series, i.e. series of the form $\sum_{n\geq 0} a_n b_n$ where $a_n$ is a Stirling number of the first kind and $\sum_{n\geq 0}b_n z^n = \phantom{}_{p+1} F_{p}(\ldots;z)$