We establish q-analogs for four congruences involving central binomial coefficients. The q-identities necessary for this purpose are shown via the q-WZ method.

For any odd prime p we obtain q-analogues of van Hamme’s and Rodriguez-Villegas’ supercongruences involving products of three binomial coefficients such as ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q²}^3\left(q^{2k}\right)/\left((-q²;q²){²}_k(-q;q){²}_{2k}²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 3 (mod 4), ${\sum }_{k=0}^{(p-1)/2}{\left[\genfrac{}{}{0pt}{}{2k}{k}\right]}_{q³}\left({(q;q³)}_k{(q²;q³)}_k{q}^{3k}\right)\left({(q⁶;q⁶)}_k²\right)\equiv 0\left(mod\left[p\right]²\right)$ for p≡ 2 (mod 3), where $\left[p\right]=1+q+\cdots +q^{p-1}$ and $(a;q)ₙ=(1-a)(1-aq)\cdots (1-a{q}^{n-1})$. We also prove q-analogues of the Sun brothers’ generalizations of the above supercongruences. Our proofs are elementary in nature and use the theory of basic hypergeometric series and combinatorial q-binomial identities including a new q-Clausen type summation formula.

We study moments of the difference $D_n\left(x\right)-x^{n}n!{\mathrm{e}}^{-1/x}$ concerning derangement polynomials $D_n\left(x\right)$. For the first moment, we obtain an explicit formula in terms of the exponential integral function and we show that it is always negative for $x>0$. For the higher moments, we obtain a multiple integral representation of the order of the moment under computation.

The object of the present work is to construct all the generalized spectral functions of a certain class of Carleman operators in the Hilbert space $L^2\left(X,\mu \right)$ and establish the corresponding expansion theorems, when the deficiency indices are (1,1). This is done by constructing the generalized resolvents of $A$ and then using the Stieltjes inversion formula.