Araştırma Makalesi
BibTex RIS Kaynak Göster

On computational formulas for parametric type polynomials and its applications

Yıl 2023, Cilt: 25 Sayı: 1, 13 - 30, 16.01.2023
https://doi.org/10.25092/baunfbed.1083754

Öz

In this paper, many formulas and identities for computing the r-parametric Hermite type polynomials are given with the help of generating functions. Using generating functions and algebraic methods, a relation is also given including these polynomials and the 2-variable Hermite Kampé de Fériet polynomials. Moreover, many relations and formulas containing the two parametric type of Apostol-Bernoulli polynomials of higher order, the two parametric type of Apostol-Euler polynomials of higher order, the two parametric type of Apostol-Genocchi polynomials of higher order and the Dickson polynomials are obtained. Finally, some special values of these polynomials and their applications with trigonometric functions are presented.

Kaynakça

  • Bretti, G. and Ricci, P. E., Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese Journal of Mathematics, 8, 3, 415-428, (2004).
  • Cesarano, C., Hermite polynomials and some generalizations on the heat equations, International Journal of Systems Applications, Engineering & Development, 8, 193-197, (2014).
  • Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G. and Torre, A., Theory of generalized Hermite polynomials, Computers & Mathematics with Applications, 28, 4, 71-83, (1994).
  • Dattoli, G., Lorenzutta, S., Maino, G., Torre, A. and Cesarano, C., Generalized Hermite polynomials and supergaussian forms, Journal of Mathematical Analysis and Applications, 203, 597-609, (1996).
  • Dere, R. and Simsek, Y., Hermite base Bernoulli type polynomials on the umbral algebra, Russian Journal of Mathematical Physics, 22, 1, 1-5, (2015).
  • Gould, W. and Hopper, A. T., Operational formulas connected with two generalizations of Hermite polynomials, Duke Mathematical Journal, 29, 51-62, (1962).
  • Hwang, K.-W. and Ryoo, C. S., Differential equations associated with two variable degenerate Hermite polynomials, Mathematics, 8, 228, (2020).
  • Kilar, N. and Simsek, Y., Computational formulas and identities for new classes of Hermite-based Milne-Thomson type polynomials: Analysis of generating functions with Euler's formula, Mathematical Methods in the Applied Sciences, 44, 6731-6762, (2021).
  • Kilar, N., Generating functions of Hermite type Milne-Thomson polynomials and their applications in computational sciences, PhD Thesis, Akdeniz University, Institute of Natural and Applied Sciences, Antalya, (2021).
  • Kurt, B., Explicit relations for the modified degenerate Apostol-type polynomials, Journal of Balıkesir University Institute of Science and Technology, 20, 2, 401-41, (2018).
  • Simsek, Y., Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus Journal of Pure and Applied Mathematics, 1, 1, 1-76, (2019).
  • Srivastava, H. M. and Choi, J., Zeta and q-Zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, (2012).
  • Srivastava, H. M. and Manocha, H. L., A treatise on generating functions, John Wiley and Sons/Ellis Horwood, New York/Chichester, (1984).
  • Srivastava, H. M., Masjed-Jamei, M. and Beyki, M. R., Some new generalizations and applications of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Rocky Mountain Journal of Mathematics, 49, 2, 681-697, (2019).
  • Dickson, L. E., The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group I, II, Annals of Mathematics, 11, 1/6, 65-120, 161-183, (1897).
  • Rassias, Th. M., Srivastava, H. M. and Yanushauskas, A., Topics in polynomials of one and several variables and their applications, Word Scientific Publishing, Singapore, (1991).
  • Ryoo, C. S., Differential equations arising from the 3-variable Hermite polynomials and computation of their zeros, in Differential Equations-Theory and Current Research, Intech Open, 81-98, (2018).
  • Masjed-Jamei, M. and Koepf, W., Symbolic computation of some power trigonometric series, Journal of Symbolic Computation, 80, 2, 273-284, (2017).
  • Kilar, N. and Simsek, Y., Relations on Bernoulli and Euler polynomials related to trigonometric functions, Advanced Studies in Contemporary Mathematics (Kyungshang), 29, 2, 191-198, (2019).
  • Kilar, N. and Simsek Y., A special approach to derive new formulas for some special numbers and polynomials, Turkish Journal of Mathematics, 44, 2217-2240, (2020).
  • Kilar, N. and Simsek, Y., Identities and relations for Hermite-based Milne-Thomson polynomials associated with Fibonacci and Chebyshev polynomials, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, RACSAM, 115, 28, 1-20, (2021).
  • Kilar, N. and Simsek, Y., New computational formulas for special numbers and polynomials derived from applying trigonometric functions to generating functions, Milan Journal of Mathematics, 89, 217-239, (2021).
  • Rainville, E. D., Special functions, The Macmillan Company, New York, (1960).
  • Srivastava, H. M., Özarslan, M. A. and Kaanoğlu, C., Some families of generating functions for a certain class of three-variable polynomials, Integral Transforms and Special Functions, 21, 12, 885-896, (2010).

Parametrik tip polinomlar için hesaplama formülleri ve uygulamaları

Yıl 2023, Cilt: 25 Sayı: 1, 13 - 30, 16.01.2023
https://doi.org/10.25092/baunfbed.1083754

Öz

Bu çalışmada, r-parametreli Hermite tipli polinomların hesaplanması için birçok formüller ve bağıntılar, üreteç fonksiyonları yardımıyla verilmiştir. Üreteç fonksiyonları ve cebirsel yöntemler kullanılarak, bu polinomları ve 2-değişkenli Hermite Kampé de Fériet polinomlarını içeren bir bağıntı da verilmiştir. Ayrıca, yüksek mertebeden iki parametreli Apostol-Bernoulli polinomları, yüksek mertebeden iki parametreli Apostol-Euler polinomları, yüksek mertebeden iki parametreli Apostol-Genocchi polinomları ve Dikson polinomlarını içeren birçok bağıntılar ve formüller elde edilmiştir. Son olarak, bu polinomların bazı özel değerleri ve trigonometrik fonksiyonlarla uygulamaları sunulmuştur.

Kaynakça

  • Bretti, G. and Ricci, P. E., Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese Journal of Mathematics, 8, 3, 415-428, (2004).
  • Cesarano, C., Hermite polynomials and some generalizations on the heat equations, International Journal of Systems Applications, Engineering & Development, 8, 193-197, (2014).
  • Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G. and Torre, A., Theory of generalized Hermite polynomials, Computers & Mathematics with Applications, 28, 4, 71-83, (1994).
  • Dattoli, G., Lorenzutta, S., Maino, G., Torre, A. and Cesarano, C., Generalized Hermite polynomials and supergaussian forms, Journal of Mathematical Analysis and Applications, 203, 597-609, (1996).
  • Dere, R. and Simsek, Y., Hermite base Bernoulli type polynomials on the umbral algebra, Russian Journal of Mathematical Physics, 22, 1, 1-5, (2015).
  • Gould, W. and Hopper, A. T., Operational formulas connected with two generalizations of Hermite polynomials, Duke Mathematical Journal, 29, 51-62, (1962).
  • Hwang, K.-W. and Ryoo, C. S., Differential equations associated with two variable degenerate Hermite polynomials, Mathematics, 8, 228, (2020).
  • Kilar, N. and Simsek, Y., Computational formulas and identities for new classes of Hermite-based Milne-Thomson type polynomials: Analysis of generating functions with Euler's formula, Mathematical Methods in the Applied Sciences, 44, 6731-6762, (2021).
  • Kilar, N., Generating functions of Hermite type Milne-Thomson polynomials and their applications in computational sciences, PhD Thesis, Akdeniz University, Institute of Natural and Applied Sciences, Antalya, (2021).
  • Kurt, B., Explicit relations for the modified degenerate Apostol-type polynomials, Journal of Balıkesir University Institute of Science and Technology, 20, 2, 401-41, (2018).
  • Simsek, Y., Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus Journal of Pure and Applied Mathematics, 1, 1, 1-76, (2019).
  • Srivastava, H. M. and Choi, J., Zeta and q-Zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, (2012).
  • Srivastava, H. M. and Manocha, H. L., A treatise on generating functions, John Wiley and Sons/Ellis Horwood, New York/Chichester, (1984).
  • Srivastava, H. M., Masjed-Jamei, M. and Beyki, M. R., Some new generalizations and applications of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Rocky Mountain Journal of Mathematics, 49, 2, 681-697, (2019).
  • Dickson, L. E., The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group I, II, Annals of Mathematics, 11, 1/6, 65-120, 161-183, (1897).
  • Rassias, Th. M., Srivastava, H. M. and Yanushauskas, A., Topics in polynomials of one and several variables and their applications, Word Scientific Publishing, Singapore, (1991).
  • Ryoo, C. S., Differential equations arising from the 3-variable Hermite polynomials and computation of their zeros, in Differential Equations-Theory and Current Research, Intech Open, 81-98, (2018).
  • Masjed-Jamei, M. and Koepf, W., Symbolic computation of some power trigonometric series, Journal of Symbolic Computation, 80, 2, 273-284, (2017).
  • Kilar, N. and Simsek, Y., Relations on Bernoulli and Euler polynomials related to trigonometric functions, Advanced Studies in Contemporary Mathematics (Kyungshang), 29, 2, 191-198, (2019).
  • Kilar, N. and Simsek Y., A special approach to derive new formulas for some special numbers and polynomials, Turkish Journal of Mathematics, 44, 2217-2240, (2020).
  • Kilar, N. and Simsek, Y., Identities and relations for Hermite-based Milne-Thomson polynomials associated with Fibonacci and Chebyshev polynomials, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, RACSAM, 115, 28, 1-20, (2021).
  • Kilar, N. and Simsek, Y., New computational formulas for special numbers and polynomials derived from applying trigonometric functions to generating functions, Milan Journal of Mathematics, 89, 217-239, (2021).
  • Rainville, E. D., Special functions, The Macmillan Company, New York, (1960).
  • Srivastava, H. M., Özarslan, M. A. and Kaanoğlu, C., Some families of generating functions for a certain class of three-variable polynomials, Integral Transforms and Special Functions, 21, 12, 885-896, (2010).
Toplam 24 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Bölüm Araştırma Makalesi
Yazarlar

Neslıhan Kılar 0000-0001-5797-6301

Yayımlanma Tarihi 16 Ocak 2023
Gönderilme Tarihi 7 Mart 2022
Yayımlandığı Sayı Yıl 2023 Cilt: 25 Sayı: 1

Kaynak Göster

APA Kılar, N. (2023). On computational formulas for parametric type polynomials and its applications. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 25(1), 13-30. https://doi.org/10.25092/baunfbed.1083754
AMA Kılar N. On computational formulas for parametric type polynomials and its applications. BAUN Fen. Bil. Enst. Dergisi. Ocak 2023;25(1):13-30. doi:10.25092/baunfbed.1083754
Chicago Kılar, Neslıhan. “On Computational Formulas for Parametric Type Polynomials and Its Applications”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 25, sy. 1 (Ocak 2023): 13-30. https://doi.org/10.25092/baunfbed.1083754.
EndNote Kılar N (01 Ocak 2023) On computational formulas for parametric type polynomials and its applications. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 25 1 13–30.
IEEE N. Kılar, “On computational formulas for parametric type polynomials and its applications”, BAUN Fen. Bil. Enst. Dergisi, c. 25, sy. 1, ss. 13–30, 2023, doi: 10.25092/baunfbed.1083754.
ISNAD Kılar, Neslıhan. “On Computational Formulas for Parametric Type Polynomials and Its Applications”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi 25/1 (Ocak 2023), 13-30. https://doi.org/10.25092/baunfbed.1083754.
JAMA Kılar N. On computational formulas for parametric type polynomials and its applications. BAUN Fen. Bil. Enst. Dergisi. 2023;25:13–30.
MLA Kılar, Neslıhan. “On Computational Formulas for Parametric Type Polynomials and Its Applications”. Balıkesir Üniversitesi Fen Bilimleri Enstitüsü Dergisi, c. 25, sy. 1, 2023, ss. 13-30, doi:10.25092/baunfbed.1083754.
Vancouver Kılar N. On computational formulas for parametric type polynomials and its applications. BAUN Fen. Bil. Enst. Dergisi. 2023;25(1):13-30.