Several properties of Racah polynomials

Yıl 2025, Cilt: 15 Sayı: 3, 809 - 818, 15.09.2025

Mustafa Dumlupınar

https://doi.org/10.17714/gumusfenbil.1660077

Öz

In this paper, bilinear and bilateral generating functions for Racah polynomials are derived, along with a theorem that provides a systematic approach for obtaining these functions. Furthermore, a new recurrence relation and an integral representation for Racah polynomials are established, enhancing their analytical framework. Special attention is given to the limiting cases of Racah polynomials, including Hahn, dual Hahn and Meixner polynomials, for which new recurrence relations are obtained. In particular, a novel integral representation for the dual Hahn polynomial is introduced, offering additional insights into its structural properties. These results contribute to the broader understanding of orthogonal polynomials, enhancing their theoretical significance and potential applications. By expanding the known properties of these polynomials, the findings may provide a basis for further mathematical research and applications in areas such as combinatorics, mathematical physics, and special functions.

Anahtar Kelimeler

Generating function , Hypergeometric function , Integral represantations , Racah polynomials , Recurrence relations

Racah polinomlarının çeşitli özellikleri

Öz

Bu makalede, Racah polinomları için bilinear ve bilateral doğurucu fonksiyonlar türetilmiş ve bu fonksiyonları elde etmek için sistematik bir yaklaşım sağlayan bir teorem sunulmuştur. Ayrıca, Racah polinomları için yeni bir rekürans bağıntısı ve integral gösterim oluşturulmuş, böylece analitik çerçeveleri zenginleştirilmiştir. Racah polinomlarının Hahn, dual Hahn ve Meixner polinomları gibi limit durumlarına özel bir ilgi gösterilmiş ve bu polinomlar için yeni rekürans bağıntıları elde edilmiştir. Özellikle, dual Hahn polinomu için yeni bir integral gösterim sunulmuş ve bu polinomun yapısal özelliklerine dair ek özellikler sağlanmıştır. Bu sonuçlar, ortogonal polinomların daha geniş bir anlayışına katkıda bulunarak teorik anlamlarını ve potansiyel uygulamalarını artırmaktadır. Bu polinomların bilinen özelliklerinin genişletilmesi, kombinatörik, matematiksel fizik ve özel fonksiyonlar gibi alanlarda daha ileri matematiksel araştırmalar ve uygulamalar için bir temel oluşturabilir.

Anahtar Kelimeler

Doğurucu fonksiyon , Hipergeometrik fonksiyon , İntegral gösterim , Racah polinomları , Rekürans bağıntıları

Kaynakça

• Agarwal, A. K., & Monocha, H. L. (1980). On some new generating functions. Matematicki Vesnik, 4(17)(32), 395–402.
• Aktaş, R., & Erkuş-Duman, E. (2015). A generalization of the extended Jacobi polynomials in two variables. Gazi University Journal of Science, 28(3), 503–521.
• Area, I., Godoy, E., Ronveaux, A., & Zarzo, A. (2004). Classical symmetric orthogonal polynomials of a discrete variable. Integral Transforms and Special Functions, 15(1), 1–12. https://doi.org/10.1080/10652460310001600672.
• Dumlupinar, M., & Erkus-Duman, E. (2023). Advances in the continuous dual Hahn polynomials. Journal of Mathematical Extension, 17(4), 1–18. https://doi.org/10.30495/JME.2023.2696.
• Erkus-Duman, E., & Choi, J. (2021). Gottlieb polynomials and their q-extensions. Mathematics, 9(13), 1499. https://doi.org/10.3390/math9131499.
• Geronimo, J. S., & Iliev, P. (2009). Bispectrality of multivariable Racah–Wilson polynomials. Constructive Approximation, 31, 417–457. https://doi.org/10.1007/s00365-009-9045-3.
• Gottlieb, M. J. (1938). Concerning some polynomials orthogonal on a finite or enumerable set of points. American Journal of Mathematics, 60(2), 453–458. https://doi.org/10.2307/2371307.
• Koekoek, R., Lesky, P. A., & Swarttouw, R. F. (2010). Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer.
• Korkmaz-Duzgun, D., & Erkus-Duman, E. (2018). The Laguerre type d-orthogonal polynomials. J. Sci. Arts, 42, 95-106.
• Ozmen, N., & Erkus-Duman, E. (2018). Some families of generating functions for the generalized Cesaro polynomials. Journal of Computational Analysis and Applications, 25, 1124–1127.
• Özmen, N. (2017). Some new properties of the Meixner polynomials. Sakarya University Journal of Science, 21(6), 1454–1462. https://doi.org/10.16984/saufenbilder.331327.
• Rahman, M. (1980). A product formula and a non-negative Poisson kernel for Racah-Wilson polynomials. Canadian Journal of Mathematics, 32(6), 1501–1517. https://doi.org/10.4153/CJM-1980-118-3.
• Rahman, M. (1981). A stochastic matrix and bilinear sums for Racah–Wilson polynomials. SIAM Journal on Mathematical Analysis, 12(2), 145–160. https://doi.org/10.1137/0512015.
• Rainville, E. D. (1960). Special Functions. New York: The Macmillan Company.
• Szegö, G. (1975). Orthogonal Polynomials (4th ed.). Rhode Island: American Mathematical Society Colloquium Publications.
• Wilson, J. A. (1977). Three-term contiguous relations and some new orthogonal polynomials. New York: Academic Press.
• Wilson, J. A. (1980). Some hypergeometric orthogonal polynomials. Society for Industrial and Applied Mathematics Journal on Mathematical Analysis, 11(4), 690–701. https://doi.org/10.1137/0511064.