On the q -Pell sequences and sums of tails

Alexander E. Patkowski

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 279-288
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Patkowski, Alexander E.. "On the $q$-Pell sequences and sums of tails." Czechoslovak Mathematical Journal 67.1 (2017): 279-288. <http://eudml.org/doc/287878>.

@article{Patkowski2017,
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.},
author = {Patkowski, Alexander E.},
journal = {Czechoslovak Mathematical Journal},
keywords = {sum of tails; $q$-series; partition; $L$-function},
language = {eng},
number = {1},
pages = {279-288},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the $q$-Pell sequences and sums of tails},
url = {http://eudml.org/doc/287878},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Patkowski, Alexander E.
TI - On the $q$-Pell sequences and sums of tails
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 279
EP - 288
AB - 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.
LA - eng
KW - sum of tails; $q$-series; partition; $L$-function
UR - http://eudml.org/doc/287878
ER -

References

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