Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 279-288, 2017

# On the $q$-Pell sequences and sums of tails

## Alexander E. Patkowski

#### Received October 13, 2015.   First published February 24, 2017.

Alexander E. Patkowski, 1390 Bumps River Rd., Centerville, Massachusetts 02632, USA, e-mail: alexpatk@hotmail.com

Abstract: We examine the $q$-Pell sequences and their applications to weighted partition theorems and values of $L$-functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous authors noted in the introduction. Showing that simple combinatorial manipulations give rise to an identity published by the present author, a weighted form of a Lebesgue partition theorem is given as the main application to partitions. The conclusion of the paper summarizes some directions for further research, pointing out that certain conditions on the $q$-polynomial would be desired, and also possibly looking at the operator equation in the present paper from the position of using modular forms.

Keywords: sum of tails; $q$-series; partition; $L$-function

Classification (MSC 2010): 11P81, 05A17

DOI: 10.21136/CMJ.2017.0550-15

Full text available as PDF.

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