Article ID: MTJPAM-D-20-00035

Title: Formulas and Relations of Special Numbers and Polynomials arising from Functional Equations of Generating Functions

Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00035; Volume 3 / Issue 1 / Year 2021, Pages 106-123

Document Type: Research Paper

Author(s): Neslihan Kilar a , Yilmaz Simsek b

aDepartment of Mathematics, Faculty of Science University of Akdeniz, TR-07058 Antalya-TURKEY

bDepartment of Mathematics, Faculty of Science University of Akdeniz, TR-07058 Antalya-TURKEY

Received: 21 October 2020, Accepted: 8 November 2020, Available online: 7 January 2021.

Corresponding Author: Neslihan Kilar (Email address:

Full Text: PDF


The aim of this paper is to introduce and investigate some new identities and formulas involving many kinds of special numbers and polynomials with help of the some known results derived from blending special formulas, generating functions and their functional equations. By using functional equations of generating functions for special numbers and polynomials, we give some relations and identities including the Genocchi polynomials of negative order, the Euler numbers and polynomials of negative order, the Changhee numbers and polynomials of negative order, the Lah numbers, the Hermite polynomials, the central factorial numbers, the Bernoulli numbers of higher order, the Daehee numbers, the Bernstein basis functions, the Stirling numbers, and also the combinatorial numbers and polynomials. Moreover, we also give several combinatorial sums and identities associated with aforementioned numbers and polynomials. Finally, we derive some finite and infinite series representations that include the incomplete gamma function and aforementioned numbers. In addition, convenient links of identities, formulas, relations and results appointed in this paper with those in earlier and future studies come to attention in detail for readers.

Keywords: Bernoulli numbers and polynomials, Euler numbers and polynomials, Genocchi numbers and polynomials, Central factorial numbers, Hermite polynomials, Stirling numbers, Lah numbers, Combinatorial numbers, Generating functions, Incomplete gamma function

  1. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publication, New York, 1970.
  2. M. Acikgoz and S. Araci, On the generating function for Bernstein polynomials, Amer. Institute of Physics Conference Proceedings CP1281, 1141-1144, 2010.
  3. S.N. Bernstein, Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités, Commun. Kharkov Math. Soc. 13, 1-2, 1912.
  4. P.L. Butzer, K. Schmidt, E.L. Stark and L. Vogt, Central factorial numbers; their main properties and some applications, Numer. Funct. Anal. Optim. 10 (5-6), 419-488, 1989, doi:10.1080/01630568908816313.
  5. L. Carlitz, Some theorems on Bernoulli numbers of higher order, Pac. Math. 2 (2), 127-139, 1952.
  6. L. Carlitz, Some formulas for the BERNOULLI and EULER polynomials, Math. Nachr. 25, 223-231, 1963.
  7. L. Carlitz, Recurrences for the Bernoulli and Euler numbers. II, Math. Nachr. 29, 151-160, 1965.
  8. M.A. Chaudhry and S.M. Zubair, Generalized incomplete Gamma functions with applications, J. Comput. Appl. Math. 55 (1), 99-123, 1994.
  9. J. Cigler, Fibonacci polynomials and central factorial numbers, Preprint,Available online:
  10. L. Comtet, Advanced Combinatorics, D. Reidel Publication Company, Dordrecht-Holland/ Boston-U.S.A., 1974.
  11. B.S. El-Desouky and A. Mustafa, New results and matrix representation for Daehee and Bernoulli numbers and polynomials, Appl. Math. Sci. 9 (73), 3593-3610, 2015 arXiv:1412.8259v1(math.CO) 29 Dec 2014.
  12. R.T. Farouki, The Bernstein polynomials basis: A centennial retrospective, Comput. Aided Geom. Des. 29, 379-419, 2012.
  13. R. Goldman, An Integrated Introduction to Computer Graphics and Geometric Modeling, CRC Press, Taylor and Francis, New York, 2009.
  14. R. Golombek, Aufgabe 1088, El Math 49, 126, 1994.
  15. D. Gun and Y. Simsek, Some new identities and inequalities for Bernoulli polynomials and numbers of higher order related to the Stirling and Catalan numbers, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 114, 167, 1-12, 2020.
  16. N. Kilar and Y. Simsek, Identities and relations for special numbers and polynomials: An approach to trigonometric functions, Filomat 34 (2), 2020.
  17. N. Kilar and Y. Simsek, Computational formulas for some classes of special numbers and polynomials: Approach to these with trigonometric and generating functions, Preprint.
  18. D.S. Kim, D.V. Dolgy, D. Kim and T. Kim, Some identities on r-central factorial numbers and r-central Bell polynomials, Adv. Difference Equ. 245, 1-11, 2019.
  19. D.S. Kim and T. Kim, Daehee numbers and polynomials, Appl. Math. Sci. (Ruse) 7 (120), 5969-5976, 2013.
  20. D.S. Kim, T. Kim, J.J. Seo and S.-H. Lee, Higher-order Changhee numbers and polynomials, Adv. Stud. Theor. Phys. 8, 365-373, 2014.
  21. D.S. Kim, T. Kim and J. Seo, A Note on Changhee numbers and polynomials, Adv. Stud. Theor. Phys. 7, 993-1003, 2013.
  22. D.S. Kim, T. Kim, S.-H. Lee and J.-J. Seo, Higher-order Daehee numbers and polynomials, Int. Journal of Math. Analysis 8, 273-283, 2014.
  23. D. Kim, Y. Simsek and J.S. So, Identities and computation formulas for combinatorial numbers including negative order Changhee polynomials, Symmetry 12 (9), 1-17, 2020, doi:10.3390/sym12010009.
  24. T. Koshy, Catalan Numbers with Applications, Oxford University Press, New York, 2009.
  25. I. Kucukoglu, B. Simsek and Y. Simsek, An approach to negative hypergeometric distribution by generating function for special numbers and polynomials, Turk. J. Math. 43, 2337-2353, 2019.
  26. I. Kucukoglu, B. Simsek and Y. Simsek, Generating functions for new families of combinatorial numbers and polynomials: Approach to Poisson Charlier polynomials and probability distribution function, Axioms 8 (4), 112, 1-16, 2019.
  27. Q-M. Luo, Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions, Taiwanese J. Math. 10, 917-925, 2006.
  28. Q-M. Luo and H.M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind, Appl. Math. Comput. 217, 5702-5728, 2011.
  29. F. Qi, X.T. Shi and F.F. Liu, Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers, Acta Univ Sapientiae. Mathematica 8 (2), 282-297, 2016.
  30. E.D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
  31. S.-H. Rim, T. Kim and S.S. Pyo, Identities between Harmonic, Hyperharmonic and Daehee numbers, J. Inequal. Appl. 168, 1-12, 2018.
  32. J. Riordan, An Introduction to Combinatorial Analysis, John Wiley Sons, Inc., New York, 1958.
  33. S. Roman, The Umbral Calculus, Dover Publications, New York, 2005.
  34. Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys. 17 (4), 495-508, 2010.
  35. Y. Simsek, Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications, Fixed Point Theory Appl. 87, 1-28, 2013.
  36. Y. Simsek, Functional equations from generating functions: A novel approach to deriving identities for the Bernstein basis functions, Fixed Point Theory Appl. 2013 (80), 1-13, 2013.
  37. Y. Simsek, Special numbers on analytic functions, Appl. Math. 5, 1091-1098, 2014.
  38. Y. Simsek, Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers, Math. Meth. Appl. Sci. 40 (7), 2347-2361, 2017.
  39. Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12, 1-35, 2018.
  40. Y. Simsek, Combinatorial identities and sums for special numbers and polynomials, Filomat 32 (20), 6869-6877, 2018.
  41. Y. Simsek, Construction of some new families of Apostol-type numbers and polynomials via Dirichlet character and p-adic q-integrals, Turk. J. Math. 42, 557-577, 2018.
  42. Y. Simsek, Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus J. Pure Appl. Math. 1 (1), 1-76, 2019.
  43. Y. Simsek, Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers, J. Math. Anal. Appl. 477, 1328-1352, 2019.
  44. Y. Simsek and M. Acikgoz, A new generating function of (q-) Bernstein-type polynomials and their interpolation function, Abstr. Appl. Anal. 2010 1-12, Article ID 769095, 2010.
  45. H.M. Srivastava, Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inf. Sci. 5 (3), 390-444, 2011.
  46. H.M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, 2012.
  47. H.M. Srivastava, I. Kucukoglu and Y. Simsek, Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials, J. Number Theory 181, 117-146, 2017.
  48. M.Z. Spivey, Combinatorial sums and finite differences, Discrete Math. 307 (24), 3130-3146, 2007.