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Topics on Meixner families

Marek Bożejko, Nizar Demni (2010)

Banach Center Publications

We shed some light on the inter-connections between different characterizations leading to the classical Meixner family. This allows us to give free analogs of both Sheffer's and Al-Salam and Chihara's characterizations in the classical case by the use of the free derivative operator. The paper closes with a discussion of the q-deformed case, |q| < 1.

Transference for hypergroups.

Giacomo Gigante (2001)

Collectanea Mathematica

A transference theorem for convolution operators is proved for certain families of one-dimensional hypergroups.

Truncations of Gauss' square exponent theorem

Ji-Cai Liu, Shan-Shan Zhao (2022)

Czechoslovak Mathematical Journal

We establish two truncations of Gauss’ square exponent theorem and a finite extension of Euler’s identity. For instance, we prove that for any positive integer n , ∑ k = 0 n ( - 1 ) k 2 n - k k ( q ; q 2 ) n - k q k + 1 2 = ∑ k = - n n ( - 1 ) k q k 2 , where n m = ∏ k = 1 m 1 - q n - k + 1 1 - q k and ( a ; q ) n = ∏ k = 0 n - 1 ( 1 - a q k ) .

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