Generalizations of the classical Laguerre polynomials and some q-analogues

We study polynomials $\{L\sbsp{n}{\alpha,M\sb0,M\sb1,\...,M\sb{N}}(x)\}\sbsp{n=0}{\infty}$ which are orthogonal with respect to the inner product(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\left\{\eqalign{&\langle f,g\rangle={1\over\Gamma(\alpha+1)} {\int\limits\sbsp{0}{\infty}}x\sp\alpha e\sp{-x}f(x)g(x)dx+ {\sum\limits\sbsp{\nu=0}{N}}M\sb\nu f\sp{(\nu)}(0)g\sp{(\nu)}(0),\cr&\alpha>-1, N\in \{0,1,2,\...\}\ {\rm and}\ M\sb\nu\ge 0\ {\rm for}\ \nu\in\{0,1,2,\...,N\}.\cr}\right.$$(TABLE/EQUATION ENDS)These polynomials are generalizations of the classical Laguerre polynomials $\{L\sbsp{n}{(\alpha)}(x)\}\sbsp{n=0}{\infty}$ since $L\sbsp{n}{\alpha,0,0,\...,0}(x)$ = $L\sbsp{n}{(\alpha)}(x)$ and of Koornwinder's generalized Laguerre polynomials $\{L\sbsp{n}{\alpha,M}(x)\}\sbsp{n=0}{\infty}$ found in (1) since $L\sbsp{n}{\alpha,M,0,0,\...,0}(x)$ = $L\sbsp{n}{\alpha,M}(x)$.

Since the inner product above cannot be obtained from any weight function in general the classical theory of orthogonal polynomials cannot be applied to derive properties of these new orthogonal polynomials $\{L\sbsp{n}{\alpha,M\sb0,M\sb1,\...,M\sb{N}}(x)\}\sbsp{n=0}{\infty}$.

In this thesis we give two definitions, an orthogonality relation and a representation as hypergeometric function for these polynomials. Moreover, we derive a second order differential equation, a recurrence relation and a Christoffel-Darboux type formula for these polynomials $\{L\sbsp{n}{\alpha,M\sb0,M\sb1,\...,M\sb{N}}(x)\}\sbsp{n=0}{\infty}$.

In some special cases some results concerning the zeros of these orthogonal polynomials are given.

Finally, a differential equation is proved for Koornwinder's generalized Laguerre polynomials $\{L\sbsp{n}{\alpha,M}(x)\}\sbsp{n=0}{\infty}$ which is of infinite order if $M>$ 0 and $\alpha>-$1 is not an integer. For $\alpha\in\{0,1,2,\...\}$ this differential equation is of finite order 2$\alpha$ + 4 provided that $M>$ 0.

The second part of this thesis deals with some q-analogues $\{L\sbsp{n}{\alpha,M\sb0,M\sb1,\...,M\sb{N}}(x;q)\}\sbsp{n=0}{\infty}$ of the polynomials $\{L\sbsp{n}{\alpha,M\sb0,M\sb1,\...,M\sb{N}}(x)\}\sbsp{n=0}{\infty}$ described in the first part. These q-analogues are generalizations of the q-Laguerre polynomials $\{L\sbsp{n}{(\alpha)}(x;q)\}\sbsp{n=0}{\infty}$ studied by Moak in (2).

For these orthogonal polynomials two definitions, two orthogonality relations and a representation as basic hypergeometric function are given. Moreover, we prove a second order q-difference equation, a recurrence relation and a Christoffel-Darboux type formula for these polynomials $\{L\sbsp{n}{\alpha,M\sb0,M\sb1,\...,M\sb{N}}(x;q)\}\sbsp{n=0}{\infty}$.

In a special case we derive some results concerning the zeros of these generalized q-Laguerre polynomials.

References. (1) T. H. Koornwinder: Orthogonal polynomials with weight function $(1-x)\sp{\alpha}(1+x)\sp{\beta}$ + $M\delta(x+1)+N\delta(x-1)$. Canadian Mathematical Bulletin 27(2), 1984, 205-214. (2) D. S. Moak: The q-analogue of the Laguerre polynomials. Journal of Mathematical Analysis and Applications 81, 1981, 20-47.