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.


References:
  [1] G. E. Andrews: $q$-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. CBMS, Reg. Conf. Ser. Math. 66, American Mathematical Society, Providence (1986). DOI 10.1090/cbms/066 | MR 0858826 | Zbl 0594.33001
  [2] G. E. Andrews: Mock theta functions. Theta functions L. Ehrenpreis, R. Gunning Bowdoin 1987, Proc. Symp. Pure Math., 49, Part 2, American Mathematical Society, Providence (1989), 283-298. DOI 10.1090/pspum/049.2/1013178 | MR 1013178 | Zbl 0678.05004
  [3] G. E. Andrews: Partitions with distinct evens. Advances in Combinatorial Mathematics. Proc. 2nd Waterloo Workshop in Computer Algebra, 2008 I. S. Kotsireas et al Springer, Berlin (2009), 31-37. DOI 10.1007/978-3-642-03562-3_2 | MR 2683225 | Zbl 1182.05008
  [4] G. E. Andrews, B. C. Berndt: Ramanujan's Lost Notebook. Part II. Springer, New York (2009). DOI 10.1007/978-1-4614-3810-6 | MR 2474043 | Zbl 1180.11001
  [5] G. E. Andrews, F. J. Dyson, D. Hickerson: Partitions and indefinite quadratic forms. Invent. Math. 91 (1988), 391-407. DOI 10.1007/BF01388778 | MR 0928489 | Zbl 0642.10012
  [6] G. E. Andrews, J. Jimenez-Urroz, K. Ono: $q$-series identities and values of certain $L$-functions. Duke Math. J. 108 (2001), 395-419. DOI 10.1215/S0012-7094-01-10831-4 | MR 1838657 | Zbl 1005.11048
  [7] K. Bringmann, B. Kane: New identities involving sums of the tails related to real quadratic fields. Ramanujan J. 23 (2010), 243-251. DOI 10.1007/s11139-009-9178-9 | MR 2739215 | Zbl 1226.11108
  [8] K. Bringmann, B. Kane: Multiplicative $q$-hypergeometric series arising from real quadratic fields. Trans. Am. Math. Soc. 363 (2011), 2191-2209. DOI 10.1090/S0002-9947-2010-05214-6 | MR 2746680 | Zbl 1228.11157
  [9] W. Y. C. Chen, K. Q. Ji: Weighted forms of Euler's theorem. J. Comb. Theory, Ser. A 114 (2007), 360-372. DOI 10.1016/j.jcta.2006.06.005 | MR 2293097 | Zbl 1110.05011
  [10] H. Cohen: $q$-identities for Maass waveforms. Invent. Math. 91 (1988), 409-422. DOI 10.1007/BF01388779 | MR 0928490 | Zbl 0642.10013
  [11] D. Corson, D. Favero, K. Liesinger, S. Zubairy: Characters and $q$-series in $\mathbb{Q}(\sqrt2)$. J. Number Theory 107 (2004), 392-405. DOI 10.1016/j.jnt.2004.03.002 | MR 2072397 | Zbl 1056.11056
  [12] N. J. Fine: Basic Hypergeometric Series and Applications. With a Foreword by George E. Andrews. Mathematical Surveys and Monographs 27, American Mathematical Society, Providence (1988). DOI 10.1090/surv/027 | MR 0956465 | Zbl 0647.05004
  [13] V. A. Lebesgue: Sommation de quelques series. J. Math. Pure. Appl. 5 (1840), 42-71.
  [14] Y. Li, H. T. Ngo, R. C. Rhoades: Renormalization and quantum modular forms, part II: Mock theta functions. Available at arXiv:1311.3044 [math.NT].
  [15] J. Lovejoy: Overpartitions and real quadratic fields. J. Number Theory 106 (2004), 178-186. DOI 10.1016/j.jnt.2003.12.014 | MR 2049600 | Zbl 1050.11085
  [16] J. Lovejoy: Overpartition pairs. Ann. Inst. Fourier (Grenoble) 56 (2006), 781-794. DOI 10.5802/aif.2199 | MR 2244229 | Zbl 1147.11061
  [17] A. E. Patkowski: A note on the rank parity function. Discrete Math. 310 (2010), 961-965. DOI 10.1016/j.disc.2009.10.001 | MR 2574849 | Zbl 1228.11158
  [18] A. E. Patkowski: An observation on the extension of Abel's lemma. Integers 10 (2010), 793-800. DOI 10.1515/integ.2010.056 | MR 2797784 | Zbl 1216.11026
  [19] A. E. Patkowski: More generating functions for values of certain $L$-functions. J. Comb. Number Theory 2 (2010), 160-170. MR 2907788 | Zbl 1319.11056
  [20] A. E. Patkowski: On curious generating functions for values of $L$-functions. Int. J. Number Theory 6 1531-1540 (2010). DOI 10.1142/S1793042110003630 | MR 2740720 | Zbl 1233.11090
  [21] J. P. O. Santos, A. V. Sills: $q$-Pell sequences and two identities ofV. A. Lebesgue. Discrete Math. 257 (2002), 125-142. DOI 10.1016/S0012-365X(01)00475-7 | MR 1931496 | Zbl 1007.05017


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://link.springer.com/journal/10587.

[Previous Article] [Contents of This Number] [Contents of Czechoslovak Mathematical Journal]