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A continued fraction of order twelve

M. Mahadeva Naika, B. Dharmendra, K. Shivashankara (2008)

Open Mathematics

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.

A note on q -partial difference equations and some applications to generating functions and q -integrals

Da-Wei Niu, Jian Cao (2019)

Czechoslovak Mathematical Journal

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 ) ...

A q -congruence for a truncated 4 ϕ 3 series

Victor J. W. Guo, Chuanan Wei (2021)

Czechoslovak Mathematical Journal

Let Φ n ( q ) denote the n th cyclotomic polynomial in q . Recently, Guo, Schlosser and Zudilin proved that for any integer n > 1 with n ≡ 1 ( mod 4 ) , ∑ k = 0 n - 1 ( q - 1 ; q 2 ) k 2 ( q - 2 ; q 4 ) k ( q 2 ; q 2 ) k 2 ( q 4 ; q 4 ) k q 6 k ≡ 0 ( mod Φ n ( q ) 2 ) , where ( a ; q ) m = ( 1 - a ) ( 1 - a q ) ⋯ ( 1 - a q m - 1 ) . In this note, we give a generalization of the above q -congruence to the modulus Φ n ( q ) 3 case. Meanwhile, we give a corresponding q -congruence modulo Φ n ( q ) 2 for n ≡ 3 ( mod 4 ) . Our proof is based on the ‘creative microscoping’ method, recently developed by Guo and Zudilin, and a 4 ϕ 3 summation formula.

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