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Certain Rodrigues Formula for F1 and ф1 Type Polynomials

Nisar KS1, Agarwal P2*, Khan MA3 and Jain S4

1Department of Mathematics, College of Arts and Science, Prince Sattam bin Abdulaziz University, Saudi Arabia

2Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India

3Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India

4Department of Mathematics, Poornima College of Engineering, Jaipur, India

*Corresponding Author:
Agarwal P
Department of Mathematics, Anand International College of Engineering, Jaipur-303012, India
Tel: +91-9694097809
E-mail:
goyal.praveen2011@gmail.com

Received: 14/10/2015 Accepted: 12/11/2015 Published: 15/11/2015

Visit for more related articles at Research & Reviews: Journal of Pure and Applied Physics

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 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. 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

image                 (1)

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

image                (2)

image                 (3)

image                 (4)

image                 (5)

image                 (6)

image                 (7)

where n and r are denote positive integers and

image                 (8)

Further,

image                 (9)

image                 (10)

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

The Appell’s F1 function defined by

image                 (11)

and the confluent hypergeometric function φ1 is defined by

image                 (12)

Rodrigues Type Formula

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

image

image                 (13)

image

image                (14)

image

image                 (15)

image

image                  (16)

image

image                 (17)

image                 (18)

image                  (19)

image                 (20)

Proof of 13:

image

image

image

image

Proof of 16:

image

image

image

image

image

Proof of 17:

image

image

image

image

Proof of 18:

image

image

image

image

Proof of 19:

image

image

image

image

Proof of 20:

image

image

image

image

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