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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 -beta integrals. At last, we derive ...
Let denote the th cyclotomic polynomial in . Recently, Guo, Schlosser and Zudilin proved that for any integer with ,
where . In this note, we give a generalization of the above -congruence to the modulus case. Meanwhile, we give a corresponding -congruence modulo for . Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a summation formula.
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