Fibonomial and Lucanomial sums through well-poised $q$-series

Wenchang Chu; Emrah Kılıç

doi:10.15672/hujms.1066540

Research Article

Fibonomial and Lucanomial sums through well-poised $q$-series

Year 2023, Volume: 52 Issue: 1, 62 - 72, 15.02.2023

Wenchang Chu Emrah Kılıç

https://doi.org/10.15672/hujms.1066540

Abstract

By making use of known identities of terminating well-poised $q$-series,
we shall demonstrate several remarkable summation formulae involving
products of two Fibonomial/Lucanomial coefficients or quotients
of two such coefficients over a third one.

Keywords

Gaussian coefficient, $q$-binomial coefficient, Well-poised $q$-series, Fibonomial coefficient, Lucanomial coefficient

References

[1] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935
[2] W. N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. 1 (1), 318–320, 1950.
[3] D. M. Bressoud, Almost poised basic hypergeometric series, Proc. Indian Acad. Sci. (Math. Sci.) 97 (1), 61–66, 1987.
[4] L. Carlitz, The characteristic polynomial of a certain matrix of binomial coefficients, The Fibonacci Quarterly, 3, 81–89, 1965.
[5] L. Carlitz, Some formulas of F. H. Jackson, Monatsh. Math. 73, 193–198, 1969.
[6] W. Chu, Basic almost poised hypergeometric series, Mem. Amer. Math. Soc. Vol. 642, 1998.
[7] W. Chu and E. Kılıç, Cubic sums of q-binomial coefficients and the Fibonomial coefficients, Rocky Mountain J. Math. 49 (8), 2557 - 2569, 2019.
[8] W. Chu and E. Kılıç, Quadratic sums of Gaussian q-binomial coefficients and Fibonomial coefficients, The Ramanujan Journal, 51 (2), 229-243, 2020.
[9] W. Chu and C. Y. Wang, Bilateral inversions and terminating basic hypergeometric series identities, Discrete Math. 309 (12), 3888–3904, 2009.
[10] G. Gasper and M. Rahman, Basic Hypergeometric Series (2nd ed.), Cambridge University Press, Cambridge, 2004.
[11] A. F. Horadam and B. J. M. Mahon, Pell and Pell–Lucas polynomials, Fibonacci Quart. 23 (1), 7–20, 1985.
[12] D. Jarden, Recurring sequences, Riveon Lematematika, Jerusalem, Israel, 1958.
[13] E. Kılıç, The generalized Fibonomial matrix, European J. Combin. 31 (1), 193–209, 2010.
[14] N. N. li and W. Chu, q-Derivative operator proof for a conjecture of Melham, Discrete Applied Mathematics, 177, 158–164, 2014.
[15] B. J. M. Mahon and A. F. Horadam, Inverse trigonometrical summation formulas involving Pell polynomials, Fibonacci Quart. 23 (4), 319–324, 1985.
[16] J. Seibert and P. Trojovsky, On some identities for the Fibonomial coefficients, Math. Slovaca 55, 9–19, 2005.
[17] P. Trojovsky, On some identities for the Fibonomial coefficients via generating function, Discrete Appl. Math. 155 (15), 2017–2024, 2007.

Year 2023, Volume: 52 Issue: 1, 62 - 72, 15.02.2023

Wenchang Chu Emrah Kılıç

https://doi.org/10.15672/hujms.1066540

Abstract

References

[1] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935
[2] W. N. Bailey, On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. 1 (1), 318–320, 1950.
[3] D. M. Bressoud, Almost poised basic hypergeometric series, Proc. Indian Acad. Sci. (Math. Sci.) 97 (1), 61–66, 1987.
[4] L. Carlitz, The characteristic polynomial of a certain matrix of binomial coefficients, The Fibonacci Quarterly, 3, 81–89, 1965.
[5] L. Carlitz, Some formulas of F. H. Jackson, Monatsh. Math. 73, 193–198, 1969.
[6] W. Chu, Basic almost poised hypergeometric series, Mem. Amer. Math. Soc. Vol. 642, 1998.
[7] W. Chu and E. Kılıç, Cubic sums of q-binomial coefficients and the Fibonomial coefficients, Rocky Mountain J. Math. 49 (8), 2557 - 2569, 2019.
[8] W. Chu and E. Kılıç, Quadratic sums of Gaussian q-binomial coefficients and Fibonomial coefficients, The Ramanujan Journal, 51 (2), 229-243, 2020.
[9] W. Chu and C. Y. Wang, Bilateral inversions and terminating basic hypergeometric series identities, Discrete Math. 309 (12), 3888–3904, 2009.
[10] G. Gasper and M. Rahman, Basic Hypergeometric Series (2nd ed.), Cambridge University Press, Cambridge, 2004.
[11] A. F. Horadam and B. J. M. Mahon, Pell and Pell–Lucas polynomials, Fibonacci Quart. 23 (1), 7–20, 1985.
[12] D. Jarden, Recurring sequences, Riveon Lematematika, Jerusalem, Israel, 1958.
[13] E. Kılıç, The generalized Fibonomial matrix, European J. Combin. 31 (1), 193–209, 2010.
[14] N. N. li and W. Chu, q-Derivative operator proof for a conjecture of Melham, Discrete Applied Mathematics, 177, 158–164, 2014.
[15] B. J. M. Mahon and A. F. Horadam, Inverse trigonometrical summation formulas involving Pell polynomials, Fibonacci Quart. 23 (4), 319–324, 1985.
[16] J. Seibert and P. Trojovsky, On some identities for the Fibonomial coefficients, Math. Slovaca 55, 9–19, 2005.
[17] P. Trojovsky, On some identities for the Fibonomial coefficients via generating function, Discrete Appl. Math. 155 (15), 2017–2024, 2007.

There are 17 citations in total.

Details

Primary Language	English
Subjects	Mathematical Sciences
Journal Section	Mathematics
Authors	Wenchang Chu 0000-0002-8425-212X Emrah Kılıç 0000-0003-0722-7382
Publication Date	February 15, 2023
Published in Issue	Year 2023 Volume: 52 Issue: 1

Cite

APA	Chu, W., & Kılıç, E. (2023). Fibonomial and Lucanomial sums through well-poised $q$-series. Hacettepe Journal of Mathematics and Statistics, 52(1), 62-72. https://doi.org/10.15672/hujms.1066540
AMA	Chu W, Kılıç E. Fibonomial and Lucanomial sums through well-poised $q$-series. Hacettepe Journal of Mathematics and Statistics. February 2023;52(1):62-72. doi:10.15672/hujms.1066540
Chicago	Chu, Wenchang, and Emrah Kılıç. “Fibonomial and Lucanomial Sums through Well-Poised $q$-Series”. Hacettepe Journal of Mathematics and Statistics 52, no. 1 (February 2023): 62-72. https://doi.org/10.15672/hujms.1066540.
EndNote	Chu W, Kılıç E (February 1, 2023) Fibonomial and Lucanomial sums through well-poised $q$-series. Hacettepe Journal of Mathematics and Statistics 52 1 62–72.
IEEE	W. Chu and E. Kılıç, “Fibonomial and Lucanomial sums through well-poised $q$-series”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 1, pp. 62–72, 2023, doi: 10.15672/hujms.1066540.
ISNAD	Chu, Wenchang - Kılıç, Emrah. “Fibonomial and Lucanomial Sums through Well-Poised $q$-Series”. Hacettepe Journal of Mathematics and Statistics 52/1 (February 2023), 62-72. https://doi.org/10.15672/hujms.1066540.
JAMA	Chu W, Kılıç E. Fibonomial and Lucanomial sums through well-poised $q$-series. Hacettepe Journal of Mathematics and Statistics. 2023;52:62–72.
MLA	Chu, Wenchang and Emrah Kılıç. “Fibonomial and Lucanomial Sums through Well-Poised $q$-Series”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 1, 2023, pp. 62-72, doi:10.15672/hujms.1066540.
Vancouver	Chu W, Kılıç E. Fibonomial and Lucanomial sums through well-poised $q$-series. Hacettepe Journal of Mathematics and Statistics. 2023;52(1):62-7.