# Some Umbral Calculus Presentations of the Chan-Chyan-Srivastava Polynomials and the Erkuș-Srivastava Polynomials

## DOI:

https://doi.org/10.4067/S0716-09172014000100006## Keywords:

Lagrange polynomials, Hermite-Kampe de Feriet polynomials, Lagrange-Hermite polynomials, Chan-ChyanSrivastava polynomials, Erkus-Srivastava polynomials, Umbral calculus, Pochhammer symbol, Multinomial theorem and multinomial coefficients.## Abstract

*In their recent investigation involving differential operators for the generalized Lagrange polynomials, Chan et. al.*[3]

*encountered and proved a certain summation identity and several other results for the Lagrange polynomials in several variables, which are popularly known in the literature as the Chan-Chyan-Srivastava polynomials. These multivariable polynomials have been studied systematically and extensively in the literature ever since then (see, for example,*[1], [4], [9], [11], [12]

*and*[13]).

*In the present paper, we investigate umbral calculus presentations ofthe Chan-Chyan-Srivastava polynomials and also of their substantially more general form, the Erkus-Srivastava polynomials*[9].

*Some other closely-related results are also considered.*

## References

[1] A. Altin and E. Erkus, On a multivariable extension of the LagrangeHermite polynomials, Integral Transforms Spec. Funct. 17, pp. 239—244, (2006).

[2] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Company, New York, (1985).

[3] W.-C. C. Chan, C.-J. Chyan and H. M. Srivastava, The Lagrange polynomials in several variables, Integral Transforms Spec. Funct. 12, pp. 139—148, (2001).

[4] K.-Y. Chen, S.-J. Liu and H. M. Srivastava, Some new results for the Lagrange polynomials in several variables, ANZIAM J. 49, pp. 243—258, (2007).

[5] G. Dattoil, P. E. Ricci and C. Cesarano, Monumbral polynomials and the associated formalism, Integral Transforms Spec. Funct. 13, pp. 155—162, (2002).

[6] G. Dattoil, P. E. Ricci and C. Cesarano, The Lagrange polynomials, the associated generalizations, and the umbral calculus, Integral Transforms Spec. Funct. 14, pp. 181—186, (2003).

[7] G. Dattoil, P. E. Ricci and C. Cesarano, Beyond the monomiality: The monumbrality principle, J. Comput. Anal. Appl. 6, pp. 77—83, (2004).

[8] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto and London, (1955).

[9] E. Erkus and H. M. Srivastava, A unified presentation of some families of multivariable polynomials, Integral Transforms Spec. Funct. 17, pp. 267—273, (2006).

[10] M. A. Khan and A. K. Shukla, On Lagrange’s polynomials of three variables, Proyecciones J. Math. 17, pp. 227—235, (1998).

[11] S.-J. Liu, Bilateral generating functions for the Lagrange polynomials and the Lauricella functions, Integral Transforms Spec. Funct. 20, pp. 519—527, (2009).

[12] S.-J. Liu, C.-J. Chyan, H.-C. Lu and H. M. Srivastava, Bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the generalized Lauricella functions, Integral Transforms Spec. Funct. 23, pp. 539—549, (2012).

[13] S.-J. Liu, S.-D. Lin, H. M. Srivastava and M.-M. Wong, Bilateral generating functions for the Erkus-Srivastava polynomials and the generalized Lauricella functions, Appl. Math. Comput. 218, pp. 7685—7693, (2012).

[14] S. Roman, The Umbral Calculus, Academic Press, New York and London, (1984).

[15] 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).

[2] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Company, New York, (1985).

[3] W.-C. C. Chan, C.-J. Chyan and H. M. Srivastava, The Lagrange polynomials in several variables, Integral Transforms Spec. Funct. 12, pp. 139—148, (2001).

[4] K.-Y. Chen, S.-J. Liu and H. M. Srivastava, Some new results for the Lagrange polynomials in several variables, ANZIAM J. 49, pp. 243—258, (2007).

[5] G. Dattoil, P. E. Ricci and C. Cesarano, Monumbral polynomials and the associated formalism, Integral Transforms Spec. Funct. 13, pp. 155—162, (2002).

[6] G. Dattoil, P. E. Ricci and C. Cesarano, The Lagrange polynomials, the associated generalizations, and the umbral calculus, Integral Transforms Spec. Funct. 14, pp. 181—186, (2003).

[7] G. Dattoil, P. E. Ricci and C. Cesarano, Beyond the monomiality: The monumbrality principle, J. Comput. Anal. Appl. 6, pp. 77—83, (2004).

[8] A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. III, McGraw-Hill Book Company, New York, Toronto and London, (1955).

[9] E. Erkus and H. M. Srivastava, A unified presentation of some families of multivariable polynomials, Integral Transforms Spec. Funct. 17, pp. 267—273, (2006).

[10] M. A. Khan and A. K. Shukla, On Lagrange’s polynomials of three variables, Proyecciones J. Math. 17, pp. 227—235, (1998).

[11] S.-J. Liu, Bilateral generating functions for the Lagrange polynomials and the Lauricella functions, Integral Transforms Spec. Funct. 20, pp. 519—527, (2009).

[12] S.-J. Liu, C.-J. Chyan, H.-C. Lu and H. M. Srivastava, Bilateral generating functions for the Chan-Chyan-Srivastava polynomials and the generalized Lauricella functions, Integral Transforms Spec. Funct. 23, pp. 539—549, (2012).

[13] S.-J. Liu, S.-D. Lin, H. M. Srivastava and M.-M. Wong, Bilateral generating functions for the Erkus-Srivastava polynomials and the generalized Lauricella functions, Appl. Math. Comput. 218, pp. 7685—7693, (2012).

[14] S. Roman, The Umbral Calculus, Academic Press, New York and London, (1984).

[15] 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).

## Published

2017-03-23

## How to Cite

[1]

H. M. Srivastava, K. S. Nisar, and M. Ahmad Khan, “Some Umbral Calculus Presentations of the Chan-Chyan-Srivastava Polynomials and the Erkuș-Srivastava Polynomials”,

*Proyecciones (Antofagasta, On line)*, vol. 33, no. 1, pp. 77-90, Mar. 2017.## Issue

## Section

Artículos