DEDUCING INFINITELY MANY MUTUALLY RELATED SEQUENCES OF ORTHOGONAL POLYNOMIALS FROM SOLUTIONS OF 1ST-ORDER INITIAL-VALUE PROBLEMS - AN ILLUSTRATIVE EXAMPLE

[摘要] When one formally solves the initial-value problem y' = {x -(alpha-x + 1)y}/x2, y(0) = 0, alpha is-an-element-of]-1, + infinity[, by inserting into the differential equation an infinite series of positive integer powers of x, one obtains an asymptotic expansion (around x = 0) of the solution y(x). The corresponding Stieltjes continued fraction can be transformed into a Jacobi continued fraction whose convergents have as denominators the generalized Laguerre polynomials and as numerators the corresponding (first) associated polynomials of the second kind, both proportionally reduced to monic polynomials. In the above-mentioned infinite Stieltjes continued fraction development of y(x), all the elements can be shifted upward over one position and in this manner one obtains the development of y1(x) := (x/y(x))-1 which is the solution of a nonlinear initial-value problem. To y1(x), there also belongs a sequence of monic orthogonal polynomials {p1,n(x)\n is-an-element-of N}. This process may be repeated an unlimited number of times. In this way one finds infinitely many related sequences of monic orthogonal polynomials {p(k,n)(x)\n is-an-element-of N}, FOR-ALL k is-an-element-of N0, all satisfying Favard's conditions as a consequence of the elements in the Stieltjes continued fraction expansion of y(x) being positive. These sets can be expressed in terms of the generalized Laguerre polynomials and their associated polynomials of the second kind of all orders. For all the sets involved, the recurrence formula, the weight function and its moments, as well as the orthogonality relation are obtained. Each sequence {p(k,n)(x)\n is-an-element-of N} can be regarded as stemming from a first-order initial-value problem y'k = f(k)(x, y(k)), y(k)(0) = 0, with known f(k)-function, and via the associated function of the second kind with subscript zero q(k),0(x), a Stieltjes-type integral representation of y(k)(x) is found.

[发布日期] 1991-06-26

[效力级别]  Proceedings Paper

[关键词] GENERALIZED LAGUERRE POLYNOMIALS;1ST-ORDER INITIAL-VALUE PROBLEMS;STIELTJES CONTINUED FRACTION DEVELOPMENTS