Certain Rodrigues Formula for F1 and ф1 Type Polynomials

Abstract

Suitable manipulations in Rodrigues type formulae can lead to numerous other type of formulae which are significatly important and are expected to find applications in probability theory and boundary value problems. Using Leibnitz formulae for nth derivative of the product of two and three functions some elegant Rodrigues type formulae for polynomials corresponding to ф1 function and Appell’s F1 function have been derived.

Keywords

Rodrigues formula, Polynomials, Arbitrary, Probability theory, Value problems, Manipulations, Special functions.

Introduction

Using Leibnitz formulae for n^th derivative of the product of two and three functions some elegant Rodrigues type formulae for polynomials corresponding to ф₁ function and Appell’s F₁ function have been derived. In recent years, many authors have been consider the ф₁ function and Appell’s F₁ type function in their studies [1-6]. In deriving the operational representations of various polynomials, we use the following fact that

                 (1)

Where λ and μ , λ ≥ μ are arbitrary real numbers. In particular, use has been made of the following results [7-12]

                (2)

                 (3)

                 (4)

                 (5)

                 (6)

                 (7)

where n and r are denote positive integers and

                 (8)

Further,

                 (9)

                 (10)

we also need the following definitions [4,5,13,14].

The Appell’s F₁ function defined by

                (11)

and the confluent hypergeometric function φ₁ is defined by

                 (12)

Rodrigues Type Formula

In this section we developed Rodrigues type formulae for F₁ and ф₁ functions

                 (13)

                (14)

                 (15)

                 (16)

                 (17)

                 (18)

                 (19)

                (20)

Proof of 13:

Proof of 16:

Proof of 17:

Proof of 18:

Proof of 19:

Proof of 20:

Similar way one can prove all the equations.

Conclusion

We conclude this paper by noting that, the results deduced above are significant and can lead to yield numerous other Rodrigues type formulae involving various special functions by suitable manipulations. More importantly, they are expected to find some applications in probability theory and boundary value problems.

References

1. Agarwal P and Jain S. Further results on fractional calculus of Srivastava polynomials, Bull. Math. Anal. Appl. 2011; 3: 167-174.
2. Agarwal P. On new unified integrals involving Appell series, Advances in Mechanical Engineering and its Applications.2012; 2: 115-120.
3. Agarwal P. On applications of fractional calculus involving summations of series, American Journal of Mathematics and Statistics. 2012; 2: 221-225.
4. AnrewsGE et al. Special Functions. Cambridge  University Press. 1999.
5. Chihara TS. An introdution toorthogonal polynomials, Gordon and Breach, New York, 1978.
6. Junesang, C. and Agarwal, P., Certain generating functions involving Appell series, Far East J. Appl. Math.  2014; 84: 25-32.
7. Khan MA and NisarKS. Bilinear and bilateral summation formulae of certain polynomials in the form of operator representations.ThaiJ. Math.  2011; 9: 673-684.
8. Khan MA andShukla AK.On binomial and trinomial operator representations of certain polynomials., Ital. J. Pure Appl. Math.  2009; 25: 103-110.
9. Khan MA and NisarKS. On binomial operator representations of certain polynomials. Ital. J. Pure Appl. Math.  2013; 31: 15-20.
10. Khan MA and NisarKS. On binomial operator representations of polynomials.Proyeccions, Chili. 2011; 30.
11. Khan MA and Nisar KS. On bilinear operator representations of Laguerre and Jacobi polynomials. International Transactions in Applied Sciences. 2009; 1: 521-526.
12. Khan MA and NisarKS. A note on binomial and trinomial operator representations of certain polynomials. Int.Journal of Math. Analysis. 2011; 5: 667-674
13. Rainville ED. Special Functions, MacMillan, New York (1960), Reprinted by Chelsea Publishing Company, Bronx, New York. 1971.
14. Srivastava HM and Manocha HL. A Treatise on Generating Functions, Ellis Horwood Limited, Chichester. 1984.