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Compositions of integers and Fibonacci numbers

Yıl 2024, Cilt: 73 Sayı: 1, 178 - 191, 16.03.2024

Busra Al Mustafa Alkan

https://doi.org/10.31801/cfsuasmas.1144430

Öz

In this paper, we deal with the compositions of the integers. We present the decompositions for both the composition sets and the odd composition sets of the integers. Thus the decompositions provide us to have not only an alternative proof of some well known identies but also many new identities for Fibonacci numbers and Lucas numbers. Thus we investigate the generating functions for the product sum of the odd composition sets of the integers and attain some functional equations.

Anahtar Kelimeler

Compositions of integers the odd combination set Fibonacci numbers Lucas numbers generating function

Destekleyen Kurum

Supported by the Scientific Research Project Administration of Akdeniz University

Proje Numarası

Research Project-5006

Kaynakça

  • Agarwal, A. K., n-Colour composition, Indian J. Pure Appl. Math., 31(11) (2000), 1421-1427.
  • Agarwal, A. K., Andrews, G. E., Rogers-Ramanujan identities for partitions with “N copies of N”, J. Combin. Theory Ser. A., 45(1) (1987), 40-49.
  • Al, B., Alkan, M., Some Relations Between Partitions and Fibonacci Numbers, In: Proceedings Book of the 2nd Mediterranean International Conference of Pure & Applied Mathematics and Related Areas (MICOPAM 2019) (Ed. by Y. Simsek, A. Bayad, M. Alkan, I. Kucukoglu and O. Ones), Antalya, Turkey, August 28-31, 2019, 14-17; ISBN: 978-2-491766-00-9.
  • Al, B., Alkan, M., On relations for the partitions of numbers, Filomat, 34(2) (2020), 567–574. DOI:10.2298/FIL2002567A
  • Al, B., Alkan, M., A Note on the Composition of a Positive Integer whose Parts are Odd Integers, International Conference on Artificial Intelligence and Applied Mathematics in Engineering Abstract Book (2022), 141. https://icaiame.com/wpcontent/uploads/2022/06/ICAIAME-2022-Accepted-Abstracts-E-Book.pdf
  • Al, B., Alkan, M., A Note on Color Compositions and the Patterns, In: Proceedings Book of the 5th Mediterranean International Conference of Pure & Applied Mathematics and Related Areas (MICOPAM 2022), 2022, 158-161. ISBN: 978-625-00-0917-8
  • Andrews, G. E., The Theory of Partitions, Addison-Wesley Publishing, New York, 1976.
  • Andrews, G. E., Erikson, K., Integer Partitions, Cambridge University Press, Cambridge, 2004.
  • Andrews, G. E., Hirschhorn, M. D., Sellers, J. A., Arithmetic properties of partitions with even parts distinct, Ramanujan Journal, 23(1–3) (2010), 169–181. DOI:10.1007/s11139-009-9158-0
  • Apostol, T. M., On the Lerch Zeta function, Pacific J. Math., 1 (1951), 161–167. DOI:10.2140/pjm.1951.1.161
  • Apostol, T. M., Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  • Archibald, M., Blecher, A., Knopfmacher, A., Inversions and parity in compositions of integers, Journal of Integer Sequences, 23 (2020). https://cs.uwaterloo.ca/journals/JIS/VOL23/Archibald/arch3.pdf
  • Birmajer, D., Gil, J. B., Weiner, M. M. D., (an + b)-color compositions, arXiv:1707.07798.
  • Chen, S. C., On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943. DOI:10.1016/j.disc.2011.02.025
  • Euler, L., Introduction to Analysis of the Infinite, Vol. 1, Springer-Verlag, 1988.
  • Ewell, J. A., Recurrences for the partition function and its relatives, Rocky Mountain Journal of Mathematics, 34(2) (2004). DOI:10.1216/rmjm/1181069871
  • Gessel I. M., Li, J., Compositions and Fibonacci identities, Journal of Integer Sequences, 16 (2013). DOI:10.48550/arXiv.1303.1366
  • Gil, B., Tomosko, J. A., Fibonacci colored compositions and applications, arXiv:2108.06462.
  • Gupta, H., Partitions - A Survey, Journal of Research of the Notional Bureau of Standards-B. Mathematical Sciences, 74B(1) (1970).
  • Heubach, S., Mansour, T., Compositions of n with parts in a set, Congr. Numer., 168 (2004), 127–143.
  • Heubach, S., Mansour, T., Combinatorics of Compositions and Words, CRC Press, 2010.
  • Hoggatt, V. E., Lind, D. A., Fibonacci and binomial properties of weighted compositions, J. Combin. Theory., 4 (1968), 121-124. DOI:10.1016/S0021-9800(68)80037-7
  • Horadam, A. F., Jacobsthal representation numbers, Fibonacci Quarterly, 34(1) (1996), 40-54.
  • Janjic, M., Some formulas for numbers of restricted words, Journal of Integer Sequences, 20 (2017).
  • Koshy, T., Fibonacci and Lucas Numbers with Applications, Canada: Wiley-Interscience Publication, 2001, 6-38.
  • Merzouka, H., Boussayoudb, A., Chelgham, M., Generating functions of generalized Tribonacci and Tricobsthal polynomials, Montes Taurus J. Pure Appl. Math., 2(2), (2020), 7–37.
  • Shapcott, C., C-color compositions and palindromes, Fibonacci Quart., 50(4) (2012), 297-303.
  • Stanley, R. P., Enumerative Combinatorics, Vol 1, 2nd edition, Cambridge University Press, 2011.
  • Simsek, Y., Generating functions for finite sums involving higher powers of binomial coeffients: Analysis of hypergeometric functions inculudinf new families of polynomilies and numbers, J.Math. Anal Appl., 477 (2019), 2328-1352.
  • Ozdemir, G., Simsek, Y., Milovanovic, G. V., Generating functions for special polynomials and numbers including Apostos-Type and Humbert-Type polynomials, Mediterr. J. Math., 14(117) (2017). DOI:10.1007/s00009-017-0918-6
  • Wilf, H. S., Generating Functionology, Academic Press, Inc., 1994.

Yıl 2024, Cilt: 73 Sayı: 1, 178 - 191, 16.03.2024

Busra Al Mustafa Alkan

https://doi.org/10.31801/cfsuasmas.1144430

Öz

Destekleyen Kurum

Akdeniz University

Proje Numarası

Research Project-5006

Kaynakça

  • Agarwal, A. K., n-Colour composition, Indian J. Pure Appl. Math., 31(11) (2000), 1421-1427.
  • Agarwal, A. K., Andrews, G. E., Rogers-Ramanujan identities for partitions with “N copies of N”, J. Combin. Theory Ser. A., 45(1) (1987), 40-49.
  • Al, B., Alkan, M., Some Relations Between Partitions and Fibonacci Numbers, In: Proceedings Book of the 2nd Mediterranean International Conference of Pure & Applied Mathematics and Related Areas (MICOPAM 2019) (Ed. by Y. Simsek, A. Bayad, M. Alkan, I. Kucukoglu and O. Ones), Antalya, Turkey, August 28-31, 2019, 14-17; ISBN: 978-2-491766-00-9.
  • Al, B., Alkan, M., On relations for the partitions of numbers, Filomat, 34(2) (2020), 567–574. DOI:10.2298/FIL2002567A
  • Al, B., Alkan, M., A Note on the Composition of a Positive Integer whose Parts are Odd Integers, International Conference on Artificial Intelligence and Applied Mathematics in Engineering Abstract Book (2022), 141. https://icaiame.com/wpcontent/uploads/2022/06/ICAIAME-2022-Accepted-Abstracts-E-Book.pdf
  • Al, B., Alkan, M., A Note on Color Compositions and the Patterns, In: Proceedings Book of the 5th Mediterranean International Conference of Pure & Applied Mathematics and Related Areas (MICOPAM 2022), 2022, 158-161. ISBN: 978-625-00-0917-8
  • Andrews, G. E., The Theory of Partitions, Addison-Wesley Publishing, New York, 1976.
  • Andrews, G. E., Erikson, K., Integer Partitions, Cambridge University Press, Cambridge, 2004.
  • Andrews, G. E., Hirschhorn, M. D., Sellers, J. A., Arithmetic properties of partitions with even parts distinct, Ramanujan Journal, 23(1–3) (2010), 169–181. DOI:10.1007/s11139-009-9158-0
  • Apostol, T. M., On the Lerch Zeta function, Pacific J. Math., 1 (1951), 161–167. DOI:10.2140/pjm.1951.1.161
  • Apostol, T. M., Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976.
  • Archibald, M., Blecher, A., Knopfmacher, A., Inversions and parity in compositions of integers, Journal of Integer Sequences, 23 (2020). https://cs.uwaterloo.ca/journals/JIS/VOL23/Archibald/arch3.pdf
  • Birmajer, D., Gil, J. B., Weiner, M. M. D., (an + b)-color compositions, arXiv:1707.07798.
  • Chen, S. C., On the number of partitions with distinct even parts, Discrete Math., 311 (2011), 940-943. DOI:10.1016/j.disc.2011.02.025
  • Euler, L., Introduction to Analysis of the Infinite, Vol. 1, Springer-Verlag, 1988.
  • Ewell, J. A., Recurrences for the partition function and its relatives, Rocky Mountain Journal of Mathematics, 34(2) (2004). DOI:10.1216/rmjm/1181069871
  • Gessel I. M., Li, J., Compositions and Fibonacci identities, Journal of Integer Sequences, 16 (2013). DOI:10.48550/arXiv.1303.1366
  • Gil, B., Tomosko, J. A., Fibonacci colored compositions and applications, arXiv:2108.06462.
  • Gupta, H., Partitions - A Survey, Journal of Research of the Notional Bureau of Standards-B. Mathematical Sciences, 74B(1) (1970).
  • Heubach, S., Mansour, T., Compositions of n with parts in a set, Congr. Numer., 168 (2004), 127–143.
  • Heubach, S., Mansour, T., Combinatorics of Compositions and Words, CRC Press, 2010.
  • Hoggatt, V. E., Lind, D. A., Fibonacci and binomial properties of weighted compositions, J. Combin. Theory., 4 (1968), 121-124. DOI:10.1016/S0021-9800(68)80037-7
  • Horadam, A. F., Jacobsthal representation numbers, Fibonacci Quarterly, 34(1) (1996), 40-54.
  • Janjic, M., Some formulas for numbers of restricted words, Journal of Integer Sequences, 20 (2017).
  • Koshy, T., Fibonacci and Lucas Numbers with Applications, Canada: Wiley-Interscience Publication, 2001, 6-38.
  • Merzouka, H., Boussayoudb, A., Chelgham, M., Generating functions of generalized Tribonacci and Tricobsthal polynomials, Montes Taurus J. Pure Appl. Math., 2(2), (2020), 7–37.
  • Shapcott, C., C-color compositions and palindromes, Fibonacci Quart., 50(4) (2012), 297-303.
  • Stanley, R. P., Enumerative Combinatorics, Vol 1, 2nd edition, Cambridge University Press, 2011.
  • Simsek, Y., Generating functions for finite sums involving higher powers of binomial coeffients: Analysis of hypergeometric functions inculudinf new families of polynomilies and numbers, J.Math. Anal Appl., 477 (2019), 2328-1352.
  • Ozdemir, G., Simsek, Y., Milovanovic, G. V., Generating functions for special polynomials and numbers including Apostos-Type and Humbert-Type polynomials, Mediterr. J. Math., 14(117) (2017). DOI:10.1007/s00009-017-0918-6
  • Wilf, H. S., Generating Functionology, Academic Press, Inc., 1994.
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Research Article
Yazarlar

Busra Al 0000-0002-1637-5355

Mustafa Alkan 0000-0002-4452-4442

Proje Numarası Research Project-5006
Yayımlanma Tarihi 16 Mart 2024
Gönderilme Tarihi 17 Temmuz 2022
Kabul Tarihi 15 Eylül 2023
Yayımlandığı Sayı Yıl 2024 Cilt: 73 Sayı: 1

Kaynak Göster

APA Al, B., & Alkan, M. (2024). Compositions of integers and Fibonacci numbers. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 73(1), 178-191. https://doi.org/10.31801/cfsuasmas.1144430
AMA Al B, Alkan M. Compositions of integers and Fibonacci numbers. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Mart 2024;73(1):178-191. doi:10.31801/cfsuasmas.1144430
Chicago Al, Busra, ve Mustafa Alkan. “Compositions of Integers and Fibonacci Numbers”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73, sy. 1 (Mart 2024): 178-91. https://doi.org/10.31801/cfsuasmas.1144430.
EndNote Al B, Alkan M (01 Mart 2024) Compositions of integers and Fibonacci numbers. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73 1 178–191.
IEEE B. Al ve M. Alkan, “Compositions of integers and Fibonacci numbers”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 73, sy. 1, ss. 178–191, 2024, doi: 10.31801/cfsuasmas.1144430.
ISNAD Al, Busra - Alkan, Mustafa. “Compositions of Integers and Fibonacci Numbers”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 73/1 (Mart 2024), 178-191. https://doi.org/10.31801/cfsuasmas.1144430.
JAMA Al B, Alkan M. Compositions of integers and Fibonacci numbers. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73:178–191.
MLA Al, Busra ve Mustafa Alkan. “Compositions of Integers and Fibonacci Numbers”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 73, sy. 1, 2024, ss. 178-91, doi:10.31801/cfsuasmas.1144430.
Vancouver Al B, Alkan M. Compositions of integers and Fibonacci numbers. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2024;73(1):178-91.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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