# Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATHEMATICS SEMESTER-IV Paper II-Advanced Special Function-II Syllabus

## Pandit S.N. Shukla University, Shahdol (M.P) M.Sc. MATHEMATICS SEMESTER-IV Paper II-Advanced Special Function-II Syllabus

Theory Paper : Max. Marks- 42

Internal Assessment: Max. Marks- 08

Note: The question paper will consist of three sections A, B&C. Section A will consist of objeclive type questions each carrving 1 marks, section B will consist of 5 short answer type questions cach earrying 3 marks and section C will consist of 2 long answer type questions each carrying 10 marks. Section B & C will have internal choice.

Unit 1- Bessel Function and Legendre polynomials: Definition of J(Z). Bessel's differential equation, Generating function, Bessel's integral with index half and an odd integer.

Unit 2- Generating function for Legendre polynomials, Rodrigues formula, Bateman's generating function, Additional generating functions. Hypergeometric forms of P.(X).

Unit 3- Special properties of Pa(X). Some more generating functions, Laplace's first integral form, Orthogonality.

Unit 4- Definition of Hermite polynomials H-(X). Pure recurrence relations, Differential recurrence relations, Rodrigue's formula, Other generating functions, Orthogonality, Expansion of polynomials, more generating functions.

Unit 5- The Laguerre Polynomials L-(X). Generating functions, Pure recurrence relations, Differential recurrence relation, Rodrigue's formula, Orthogonality, Expansion of polynomials, special properties, Other generating functions.

Recommended Books:

[1]E.D. Rainville, Special Functions, The Macmillan Co., New York, 1971.

[2] H.M.Srivastava, K.C.Gupta and S.P.Goyal, The H-function of One and Two variables with applications, South Asian Publication, New Delhi,

[3] N . Saran, S.D.Sharma and Trivedi, Special Functions with applications, Pragati Prakashan, 1986.

[4] N.N.Lebdev, Special functions and their applications, Prentice Hall, Englewood Cliffs, New Jersey, USA, 1965.