Let ${\Phi }_n\left(q\right)$ denote the $n$th cyclotomic polynomial in $q$. Recently, Guo, Schlosser and Zudilin proved that for any integer $n>1$ with $n\equiv 1(mod4)$, $$\sum _{k=0}^{n-1}\frac{{(q^{-1};q^2)}_k^2{(q^{-2};q^4)}_k}{{(q^2;q^2)}_k^2{(q^4;q^4)}_k}{q}^{6k}\equiv 0(mod{\Phi }_n{\left(q\right)}^2),$$ where ${(a;q)}_m=(1-a)(1-aq)\cdots (1-a{q}^{m-1})$. In this note, we give a generalization of the above $q$-congruence to the modulus ${\Phi }_n{\left(q\right)}^3$ case. Meanwhile, we give a corresponding $q$-congruence modulo ${\Phi }_n{\left(q\right)}^2$ for $n\equiv 3(mod4)$. Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a ${}_4{\varphi }_3$ summation formula.

350 years ago in Spring of 1655 Sir William Brouncker on a request by John Wallis obtained a beautiful continued fraction for 4/π. Brouncker never published his proof. Many sources on the history of Mathematics claim that this proof was lost forever. In this paper we recover the original proof from Wallis' remarks presented in his Arithmetica Infinitorum. We show that Brouncker's and Wallis' formulas can be extended to MacLaurin's sinusoidal spirals via related Euler's products. We derive Ramanujan's...

The present paper deals with certain generating functions and recurrence relations for $q$-Laguerre polynomials through the use of the $T_{k,q,x}$-operator introduced in an earlier paper [7].

The group SU(1,d) acts naturally on the Hilbert space $L²\left(Bd{\mu }_{\alpha }\right)(\alpha >-1)$, where B is the unit ball of ${ℂ}^d$ and $d{\mu }_{\alpha }$ the weighted measure ${(1-|z\left|²\right)}^{\alpha }dm\left(z\right)$. It is proved that the irreducible decomposition of the space has finitely many discrete parts and a continuous part. Each discrete part corresponds to a zero of the generalized Harish-Chandra c-function in the lower half plane. The discrete parts are studied via invariant Cauchy-Riemann operators. The representations on the discrete parts are equivalent to actions on some holomorphic...

This is an indicatory presentation of main definitions and theorems of Fibonomial Calculus which is a special case of ψ-extented Rota's finite operator calculus [7].