ON A SYSTEM OF FUNCTIONS GENERATING CLASSICAL ORTHOGONAL SYSTEMS OF ALGEBRAIC POLYNOMIALS

Abstract

In contrast to the two well-known methods for introducing classical orthogonal systems of algebraic polynomials (see [1--5], respectively), this paper proposes a completely different method for introducing these same polynomials. At the same time, integral representations for them that are different from the known ones are also given. The latter, in comparison with the Rodrigues formulas known in the literature, are simple and, what is more important, the integrands are not multivalued, but single-valued. In this paper it is proved that the system of functions $$I^{(\alpha,\beta)}_{n,n_1,a}=\frac{{\ae}_{n,n_1,a}}{2\pi_i}\int\limits_C\frac{(1-\frac{x}{a})^{n_1+\zeta}(\frac{x}{a})^{-\zeta}}{\Gamma(n_1+\zeta+\alpha+1)\Gamma(-\zeta+\beta+1)}\cdot\frac{d\zeta}{\prot\limits^m_{v=0}(\zeta+v)},$$ where $n_1\geq n\geq 0$ are integers, $\alpha>-1, \beta>-1, a>0$ are arbitrary numbers, $x\in(0,a)$, the simple contour $C$ encloses neighborhoods of points $0,-1,-2,...,-n, {\ae}{n,n_1,a}=\Gamma(n_1+n+\alpha+\beta=2),$ generates the above-mentioned orthogonal polynomials, namely: for $n_1=n, a=1$ we obtain the Jacobi polynomials, and for $n_1=a\rightarrow\infty$ -- the Laguerre polynomials $L^{(\beta)}_n(x),$ multiplied by $\exp^{-x}$. At the endpoints $[0,a] \mathrm{Y}\limits^{(\alpha, \beta)_{n,n_1,a}(x)} $ is understood in the sense of $x\rightarrow0+, x\rightarrow a–$.