Q-Kummer’s Equation And Extensions Of Rodrigues Formula

The special functions of mathematical physics,namely,the classical orthogonal polynomials(Jacobi,Laguerre,and Hermite polynomials),the cylindrical and hyper-geometric functions,are solutions of the second order differential equation with the same type.These equations are called hypergeometric differential equations.With more and more special functions are introduced and studied,there are three discretiza-tions of the hypergeometric differential equations.In this thesis,we establish a q-analogue of Kummer’s equation and give extensions of Rodrigues formula in the three discretizations.In Chapter 1,hypergeometric differential equations and their normal forms are presented.Moreover,three discretizations of hypergeometric differential equations are also given.In Chapter 2,we first compute the operator form of q-hypergeometric equation,then using the operator form,the general form are established.From the general form,a q-analogue of Kummer’s equation called q-Kummer’s equation is obtained for the first time.Then,we give the notion of singular point in the case of q-difference equa-tions.Linearly independently formal series solution are obtained by using Frobenius method in ODEs.Given the formal series solutions are divergent,we have two inte-gral solutions which are convergent under certain conditions.In the end,we establish six contiguous relations about the q-hypergeometric series 1φ1.In Chapter 3,we give a derivation about the q-analogue of Rodrigues formula of the hypergeometric q-difference equations.Then,two ways of definition about adjoint equations in the case of q-difference equations are given.Using adjoint equations,we give two explicit forms of solutions except the polynomials solutions.Moreover,the extended Rodrigues formula are closely related with the second kind function of hy-pergeometric q-difference equations exemplified by the little q-Jacobi polynomials.In the end,a more general extension of Rodrigues formula in the case of hypergeometric h-difference equations is presented.Chapter 4 is devoted to the hypergeometric difference equations on non-uniform lattices.Two notions of integrations on general lattices are given in the first sec-tion.Inspired by the cases of differential equations and its q-analogue,we give a definition of adjoint equation corresponding to hypergeometric difference equations on non-uniform lattices.Then using the adjoint equation,we give an explicit form about the solutions of hypergeometric difference equations on non-uniform lattices except polynomial solutions.We also obtain a three-term recurrence relation about a special function which is a solution of the adjoint equation.In the end,a more gen-eral extension of Rodrigues formula is given by using a similar method as the one in Chapter 3.