### A bound for certain bibasic sums and applications.

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In this paper, we establish several explicit evaluations, reciprocity theorems and integral representations for a continued fraction of order twelve which are analogues to Rogers-Ramanujan’s continued fraction and Ramanujan’s cubic continued fraction.

We study the condition on expanding an analytic several variables function in terms of products of the homogeneous generalized Al-Salam-Carlitz polynomials. As applications, we deduce bilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. We also gain multilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. Moreover, we obtain generalizations of Andrews-Askey integrals and Ramanujan $q$-beta integrals. At last, we derive $U(n+1)$...

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 $n\equiv 1\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}4)$, $$\sum _{k=0}^{n-1}\frac{{({q}^{-1};{q}^{2})}_{k}^{2}{({q}^{-2};{q}^{4})}_{k}}{{({q}^{2};{q}^{2})}_{k}^{2}{({q}^{4};{q}^{4})}_{k}}{q}^{6k}\equiv 0\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}{\Phi}_{n}{\left(q\right)}^{2}),$$ 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 $n\equiv 3\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}4)$. Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a ${}_{4}{\varphi}_{3}$ summation formula.