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Year 2020, Volume: 8 Issue: 2, 123 - 129, 15.10.2020
https://doi.org/10.36753/mathenot.708004

Abstract

References

  • Reference1 Gould, H.W.: Combinatorial identities, Morgantown, W. Va., (1972).
  • Reference2 Kılıç E., Ömür, N. and Türker Ulutaş, Y.: Alternating sums of the powers of Fibonacci and Lucas numbers. Miskolc Mathematical Notes. 12 (1), 87-103 (2011).
  • Reference3 Kılıç E., Türker Ulutaş, Y., Ömür, N.: Formulas for weighted binomial sums with the powers of\ terms of binary recurrences. Miskolc Mathematical Notes. 13 (1), 53-65 (2012).
  • Reference4 Kılıç E., Ömür, N.. and Koparal, S.: On alternating weighted binomial sums with falling factorials. Bulletin of Mathematical Analysis and Applications. 9(1), 58-64 (2017).
  • Reference5 Kılıç E., Ömür, N.. and Koparal, S.: Formulae for two weighted binomial identities with the falling factorials.} Ars Combinatoria. 138 223-231 (2018).
  • Reference6 Khan M. and Kwong H.: Some binomial identities associated with the generalized natural number sequence. The Fibonacci Quarterly. 49 (1), 57-65 (2011).
  • Reference7 Kılıç E., and Taşdemir, F.:On binomial double sums with Fibonacci and Lucas Numbers-I. Ars Combinatoria. 144, 173-185 (2019).
  • Reference8 Kılıç E., and Taşdemir, F.: On binomial double sums with Fibonacci and Lucas Numbers-II. Ars Combinatoria. 144, 345-354 (2019).
  • Reference9 Ozeki, K.: On Melham's sum. Fibonacci Quarterly. 46/47 (2), 107-110 (2008/09).
  • Reference10 Prodinger, H.: On a sum of Melham and its variants. Fibonacci Quarterly. 46/47 (3), 207-215 (2008/09).
  • Reference11 Wiemann, M. and Cooper, C.: Divisibility of an F-L type convolution, in Applications of Fibonacci numbers. Ser. Proceedings of the 10th International Research Conference on Fibonacci Numbers and their Applications held at Northern Arizona University, Dorrecht:Klywer Acad. Publ. 9, 267-287 (2004).

On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials

Year 2020, Volume: 8 Issue: 2, 123 - 129, 15.10.2020
https://doi.org/10.36753/mathenot.708004

Abstract

In this paper, we consider and obtain binomial sums and alternating binomial
sums including falling factorial of the summation indices. For example, for
nonnegative integer $m,$
\begin{eqnarray*}
&&\sum\limits_{k=0}^{n}\dbinom{n}{k}k^{\underline{m}}U_{2k}^{2m}=\frac{n^{\underline{m}}}{\left( p^{2}+4\right) ^{m}}\left(
\sum\limits_{i=0}^{m}\left( -1\right) ^{i}\dbinom{2m}{i}V_{2\left(
m-i\right) }^{n-m}V_{2\left( m+n\right) \left( m-i\right) }-\left( -1\right)
^{m}2^{n-m}\dbinom{2m}{m}\right),

References

  • Reference1 Gould, H.W.: Combinatorial identities, Morgantown, W. Va., (1972).
  • Reference2 Kılıç E., Ömür, N. and Türker Ulutaş, Y.: Alternating sums of the powers of Fibonacci and Lucas numbers. Miskolc Mathematical Notes. 12 (1), 87-103 (2011).
  • Reference3 Kılıç E., Türker Ulutaş, Y., Ömür, N.: Formulas for weighted binomial sums with the powers of\ terms of binary recurrences. Miskolc Mathematical Notes. 13 (1), 53-65 (2012).
  • Reference4 Kılıç E., Ömür, N.. and Koparal, S.: On alternating weighted binomial sums with falling factorials. Bulletin of Mathematical Analysis and Applications. 9(1), 58-64 (2017).
  • Reference5 Kılıç E., Ömür, N.. and Koparal, S.: Formulae for two weighted binomial identities with the falling factorials.} Ars Combinatoria. 138 223-231 (2018).
  • Reference6 Khan M. and Kwong H.: Some binomial identities associated with the generalized natural number sequence. The Fibonacci Quarterly. 49 (1), 57-65 (2011).
  • Reference7 Kılıç E., and Taşdemir, F.:On binomial double sums with Fibonacci and Lucas Numbers-I. Ars Combinatoria. 144, 173-185 (2019).
  • Reference8 Kılıç E., and Taşdemir, F.: On binomial double sums with Fibonacci and Lucas Numbers-II. Ars Combinatoria. 144, 345-354 (2019).
  • Reference9 Ozeki, K.: On Melham's sum. Fibonacci Quarterly. 46/47 (2), 107-110 (2008/09).
  • Reference10 Prodinger, H.: On a sum of Melham and its variants. Fibonacci Quarterly. 46/47 (3), 207-215 (2008/09).
  • Reference11 Wiemann, M. and Cooper, C.: Divisibility of an F-L type convolution, in Applications of Fibonacci numbers. Ser. Proceedings of the 10th International Research Conference on Fibonacci Numbers and their Applications held at Northern Arizona University, Dorrecht:Klywer Acad. Publ. 9, 267-287 (2004).
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sibel Koparal 0000-0001-9574-9652

Neşe Ömür 0000-0002-3972-9910

Publication Date October 15, 2020
Submission Date March 23, 2020
Acceptance Date May 5, 2020
Published in Issue Year 2020 Volume: 8 Issue: 2

Cite

APA Koparal, S., & Ömür, N. (2020). On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials. Mathematical Sciences and Applications E-Notes, 8(2), 123-129. https://doi.org/10.36753/mathenot.708004
AMA Koparal S, Ömür N. On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials. Math. Sci. Appl. E-Notes. October 2020;8(2):123-129. doi:10.36753/mathenot.708004
Chicago Koparal, Sibel, and Neşe Ömür. “On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers With Falling Factorials”. Mathematical Sciences and Applications E-Notes 8, no. 2 (October 2020): 123-29. https://doi.org/10.36753/mathenot.708004.
EndNote Koparal S, Ömür N (October 1, 2020) On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials. Mathematical Sciences and Applications E-Notes 8 2 123–129.
IEEE S. Koparal and N. Ömür, “On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials”, Math. Sci. Appl. E-Notes, vol. 8, no. 2, pp. 123–129, 2020, doi: 10.36753/mathenot.708004.
ISNAD Koparal, Sibel - Ömür, Neşe. “On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers With Falling Factorials”. Mathematical Sciences and Applications E-Notes 8/2 (October 2020), 123-129. https://doi.org/10.36753/mathenot.708004.
JAMA Koparal S, Ömür N. On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials. Math. Sci. Appl. E-Notes. 2020;8:123–129.
MLA Koparal, Sibel and Neşe Ömür. “On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers With Falling Factorials”. Mathematical Sciences and Applications E-Notes, vol. 8, no. 2, 2020, pp. 123-9, doi:10.36753/mathenot.708004.
Vancouver Koparal S, Ömür N. On Binomial Sums and Alternating Binomial Sums of Generelized Fibonacci Numbers with Falling Factorials. Math. Sci. Appl. E-Notes. 2020;8(2):123-9.

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