Certain Laguerre-based Generalized Apostol Type Polynomials

Main Article Content

Junesang Choi
Nabiullah Khan
Talha Usman


A variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential applications in diverse research areas. In this paper, we aim to introduce Laguerre-based generalized Apostol type polynomials and investigate some properties and identities involving them. Among them, some differential-recursive relations for the Hermite-Laguerre polynomials, which are expressed in terms of generalized Apostol type numbers and the Laguerre-based generalized Apostol type polynomials, an implicit summation formula and addition-symmetry identities for the Laguerre-based generalized Apostol type polynomials are presented. The results presented here, being very general, are pointed out to reduce to yield some known or new formulas and identities for relatively simple polynomials and numbers.

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How to Cite
Choi, J., Khan, N., & Usman, T. (2022). Certain Laguerre-based Generalized Apostol Type Polynomials. Tamkang Journal of Mathematics, 53(1), 59–74. https://doi.org/10.5556/j.tkjm.53.2022.3491
Author Biography

Junesang Choi, Dongguk University

Full Professor


M. Abramowitz and I. A. Stegun (Editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Tenth Printing, National Bureau of Standards, Applied Mathematics Series 55, National Bureau of Standards, Washington, D.C., 1972; Reprinted by Dover Publications, New York, 1965.

E. T. Bell, Exponential polynomials, Ann. Math. 35(1934), 258–277.

Yu. A. Brychkov, On multiple sums of special functions, Integral Trans. Spec. Func. 21(12) (2010), 877–884.

J. Choi, Notes on formal manipulations of double series, Commun. Korean Math. Soc.18(4) (2003), 781–789.

L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions (Translated from the French by J. W. Nienhuys), Reidel, Dordrecht and Boston, 1974.

G. Dattoli, S. Lorenzutta and C. Cesarano, Finite sums and generalized forms of Bernoulli polynomials, Rend. Mat. 19 (1999), 385–391.

G. Dattoli and A. Torre, Operational methods and two variable Laguerre polynomials, Atti Acad. Torino 132 (1998), 1–7.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Func- tions, Vol. 3, McGraw-Hill Book Company, New York, Toronto, and London, 1955.

E. R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, New Jersey, 1975.

Y.He,S.Araci,H.M.SrivastavaandM.Abdel-Aty,Higher-order convolutions for Apostol- Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Mathematics 6(12) (2019), http://dx.doi.org/10.3390/math6120329

N. U. Khan and T. Usman, A new class of Laguerre-based generalized Apostol polynomials, Fasc. Math. 57 (2016), 67–89.

N. U. Khan and T. Usman, A new class of Laguerre-based poly-Euler and multi poly-Euler polynomials, J. Anal. Num. Theor. 4(2) (2016), 113–120.

N. U. Khan and T. Usman, A new class of Laguerre poly-Bernoulli numbers and polynomials, Adv. Stud. Contemporary Math. 27(2) (2017), 229–241.

N. U. Khan, T. Usman and A. Aman, Generating functions for Legendre-based poly- Bernoulli numbers and polynomials, Honam Math. J. 39(2) (2017), 217–231.

N. U. Khan, T. Usman and J. Choi, Certain generating function of Hermite-Bernoulli- Laguerre polynomials, Far East J. Math. Sci. 101(4) (2017), 893–908.

N. U. Khan, T. Usman and J. Choi, A new generalization of Apostol type Laguerre-Genocchi polynomials, C. R. Acad. Sci. Paris, Ser. I 355 (2017), 607–617. http://dx.doi.org/10. 1016/j.crma.2017.04.010

N. U. Khan, T. Usman and J. Choi, A new class of generalized polynomials, Turk. J. Math. 42 (2018), 1366–1379.

N. U. Khan, T. Usman and J. Choi, A new class of generalized Laguerre-Euler polynomials, RACSAM 113 (2019), 861–873. https://doi.org/10.1007/s13398-018-0518-8

N. U. Khan, T. Usman and W. A. Khan, A new class of Laguerre-based generalized Hermite-Euler polynomials and its properties, Kragujevac J. Math. (2018), in press.

D.-Q. Lu and Q.-M. Luo, Some properties of the generalized Apostol type polynomials, Bound. Value Probl. 2013(2013), Article ID 64. http://www.boundaryvalueproblems. com/content/2013/1/64

Y. L. Luke, The Special Functions and Their Approximations, Vol. I, Academic Press, New York and London, 1969.

Q.-M. Luo, Apostol Euler polynomials of higher order and Gaussian hypergeometric func- tions, Taiwanese J. Math. 10(4) (2006), 917–925.

Q.-M. Luo, q-extensions for the Apostol-Genocchi polynomials, Gen. Math. 17(2) (2009), 113–125.

Q.-M. Luo, Extensions for the Genocchi polynomials and its fourier expansions and integral representations, Osaka J. Math. 48 (2011), 291–310.

Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol- Euler polynomials, J. Math. Anal. Appl. 308(1) (2005), 290–302.

Q.-M. Luo and H. M. Srivastava, Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comput. Math. Appl. 51 (2006), 631–642.

Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Genocchi polynomials and the Stirling number of the second kind, Appl. Math. Comput. 217 (2011), 5702–5728.

W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Func- tions of Mathematical Physics, Third Enlarged edition, Die Grundlehren der Mathematis- chen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtingung der Anwen- dungsgebiete, Bd. 52, Springer-Verlag, Berlin, Heidelberg and New York, 1966.

G. Ozdemir, Y. Simsek and G. V. Milovanović, Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials, Mediterr. J. Math. 14 (2017), Article ID 17.

E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.

Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12 (2018), 001–035.

H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.

H. M. Srivastava, M. Grag and S. Choudhary, A new generalization of the Bernoulli and related polynomials, Russian J. Math. Phys. 17 (2010), 251–261.

H. M. Srivastava, M. Grag and S. Choudhary, Some new families of generalized Euler and Genocchi polynomials, Taiwanese J. Math. 15 (2011), 283–305.

H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1984.

S.-l. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308 (2008), 550–554. https://doi.org/10.1016/j.disc.2007.03.030

S. L. Yang and Z. K. Qiao, Some symmetry identities for the Euler polynomials, J. Math. Res. Exposition 30(3) (2010), 457–464.

Z. Zhang and H. Yang, Several identities for the generalized Apostol-Bernoulli polynomials, Comput. Math. Appl. 56(12) (2008), 2993–2999.