SPECIAL FUNCTIONS

Gauss hypergeometric functions: Definition  and its properties, Condition of convergence Integral representation, Gauss Theorem, Vandermonde’s Theorem, Kummer’s theorem, Linear transformation, Differentiation formulae.
 

Gauss’s hypergeometric differential equation and it’s solution, relation between the solution of hypergeometric equation, Relations of contiguity, Two summation theorems, Kummer’s confluent hypergeometric function: Definition and it’s properties, Integral representation, Differentiation, Kummer’s first transformation.
 

Legendre polynomials and functions: Definition, solution of Legendre’s equation, Legendre functions of the first and second kind, Generating functions(First formula), Rodrigues formula for Pn(x), Orthogonality of Legendre polynomials, Recurrence relations for Pn(x), Beltrami’s result, Christoffel expansion, Christoffel’s summation formula, Relation between Pn(x) and Qn(x).

Bessel Functions: Bessel equation and it’s solution, Recurrence relations, Generating function, Integral Representations of Bessel function.
 

Hermite polynomials: Definition, Generating function, Recurrence relations, Orthogonality of Hn(x), Rodrigues formula, Hermite’s differential equation and it’s solution. Laguerre polynomials: Laguerre’s differential equation and it’s solutions, Generating function, Rodrigue formula, Orthogonality and simple Laguerre  polynomials, Recurrence Relati