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Algorithms for Classical Orthogonal Polynomials Konrad-Zuse-Zentrum für Informationstechnik Berlin Takustr. 7, D-14195 Berlin - Dahlem Wolfram Ko epf Dieter Schmersau Algorithms for Classical Orthogonal Polynomials at Berlin Fachb ereich Mathematik und Informatik der Freien Universit Preprint SC Septemb er Algorithms for Classical Orthogonal Polynomials Wolfram Ko epf Dieter Schmersau koepfzibde Abstract In this article explicit formulas for the recurrence equation p x A x B p x C p x n+1 n n n n n1 and the derivative rules 0 x p x p x p x p x n n+1 n n n n1 n and 0 p x p x x p x x n n n n1 n n resp ectively which are valid for the orthogonal p olynomial solutions p x of the dierential n equation 00 0 x y x x y x y x n of hyp ergeometric typ e are develop ed that dep end only on the co ecients x and x which themselves are p olynomials wrt x of degrees not larger than and resp ectively Partial solutions of this problem had b een previously published by Tricomi and recently by Yanez Dehesa and Nikiforov Our formulas yield an algorithm with which it can b e decided whether a given holonomic recur rence equation ie one with p olynomial co ecients generates a family of classical orthogonal p olynomials and returns the corresp onding data density function interval including the stan dardization data in the armative case In a similar way explicit formulas for the co ecients of the recurrence equation and the dierence rule x rp x p x p x p x n n n+1 n n n n1 of the classical orthogonal p olynomials of a discrete variable are given that dep end only on the co ecients x and x of their dierence equation x ry x x y x y x n Here y x y x y x and ry x y x y x denote the forward and backward dierence op erators resp ectively In particular this solves the corresp onding inverse problem to nd the classical discrete orthogonal p olynomial solutions of a given holonomic recurrence equation Polynomials of the Hyp ergeometric Typ e A longstanding problem in the theory of sp ecial functions whose solution can b e very helpful in applied mathematics as well as in many quantummechanical problems of physics is the determination of the dierentiation formulas of the hyp ergeometrictyp e orthogonal p olynomials p x only from the co ecients of the dierential equation n 00 0 x y x x y x y x n which is satised by these p olynomials n y x p x k x n N f g k n n 0 n The co ecients x x and turn out to b e themselves p olynomials wrt x of degrees n not larger than and resp ectively This problem was partially solved by Tricomi Chapter IV in the sense that he was able to calculate the co ecients and of the derivative rule n n n 0 x p x x p x p x n n n n n1 n However his formula for was not only in terms of the co ecients of and k but n n furthermore the second highest co ecients of p x were involved and to evaluate he n n needed to know also the co ecients of the recurrence equation p x A x B p x C p x n+1 n n n n n1 another structural prop erty of orthogonal p olynomial systems Since the p olynomials p x given by are completely determined by the dierential equation n and their leading co ecients k n N it is desirable to obtain the recurrence equation n 0 and the derivative rule from these informations alone Recently Yanez Dehesa and Nikiforov presented such formulas which however are ad ditionally in terms of the constant D given by a representation of the typ e n Z n s s D n ds p x n n+1 x s x C 0 for p x x b eing solution of the equation and C b eing a contour satisfying n certain b oundary conditions Their development is more general in the sense that they did not assume that n is an integer On the other hand the assumption that n is an integer implies that the contour C is closed the integral representation b eing equivalent to the Ro drigues representation n d E n n x x p x n n x dx where n E D n n i and the solutions are classical orthogonal p olynomials with density x In this article we represent the co ecients of b oth and in terms of x x and the term ratio k k alone hence giving a complete solution of the prop osed problem n+1 n It is clear that our formulas should dep end additionally on the leading co ecients k since n such a standardization can b e prescrib ed arbitrarily If one takes the monic standardization ie k then the formulas in fact dep end only on the co ecients of the dierential equation n For the classical orthogonal p olynomials our formulas are stronger than Yanez Dehesas and Nikiforovs result since k is intrinsic part of p x whereas the constants D E are not n n n n Moreover we will give D and E in terms of the co ecients of the dierential equation to o n n Algebraically two identities dierential equation and recurrence equation eg are needed to deduce the third one derivative rule eg see whereas here kind of magic we would like to deduce two from one That this is p ossible is due to the analytic knowledge that orthogonal p olynomial solutions of the dierential equation satisfy some structural prop erties namely the recurrence equation and derivative rule take sp ecial forms We make the general assumption that our p olynomials p x are orthogonal wrt a measure n ie Z if m n p x p x dx n m h if m n n I where I denotes an appropriate integration path for example a real interval Ma jor to ols in our development are the following wellknown structural prop erties of such families of orthogonal p olynomials Lemma Any system of p olynomials fp x j n N g p b eing of exact degree n orthogo n 0 n nal with resp ect to a measure satises a threeterm recurrence equation of the form p x A x B p x C p x n N p x n+1 n n n n n1 0 1 A B and C not dep ending on x n n n Pro of This prop erty is wellknown see eg Chapter IV To prove it one substitutes n+1 equates the co ecients of x and gets immediately that k n+1 A n k n With this choice we study the dierence p x A x p x Since this is a p olynomial of n+1 n n degree not larger than n it can b e decomp osed as n X p x A x p x d p x n+1 n n j n j =0 We cho ose m n and multiply by p x Integrating with resp ect to yields m Z Z p x p x dx p x A x p x dx d h m n+1 m n n m m I I where on the right hand side was applied Both left hand integrals vanish since p x is m orthogonal to p x and since xp x as a p olynomial of degree not larger than n is n+1 m orthogonal to p x implying d This gives the result 2 n m The second imp ortant structural prop erty for our considerations is given by Lemma Any system of p olynomials fp x j n N g p b eing of exact degree n that n 0 n are solutions of the dierential equations and furthermore orthogonal with resp ect to a measure x x dx having weight function x satises a derivative rule of the form 0 x p x x p x p x n N f g n n n n n1 n and not dep ending on x n n n n+1 Pro of Substituting and equating the co ecients of x one gets immediately that a n n In x it is shown by an elementary argument that under the given conditions the solutions p x of the dierential equations are orthogonal with resp ect to the weight function n R (x) C dx (x) x e x given by Pearsons dierential equation d x x x x dx for a suitable constant C in a suitable interval I dep ending on the zeros of x Hence multiplying by x the dierential equation takes the selfadjoint form d 0 x x y x x y x n dx Using this identity Tricomi showed that IV Z 0 x x p x f x dx n I 0 x x for any p olynomial f x of degree n If holds then the degree of x p n n is n Hence one can write n X 0 x p x x e p x n j n n j =0 As ab ove from one can deduce that e for j n see Chapter IV 2 j An immediate consequence is the following Corollary Any system of p olynomials fp x j n N g p b eing of exact degree n that n 0 n are solutions of the dierential equation and furthermore orthogonal with resp ect to a measure x x dx having weight function x satises a derivative rule of the form 0 x p x p x p x p x n N n n+1 n n n n1 n and not dep ending on x n n n n+1 Pro of Substituting in and equating the co ecients of x one gets immediately that k n a n n k n+1 Substituting in one gets moreover 0 x p x x p x p x n n n n n1 n n p x p x p x B p x C p x n n n1 n+1 n n n n1 n A n hence is valid with B C n n n 2 n n n n n n n A A A n n n Classical Orthogonal Polynomials of an Interval In this section we give the prop osed explicit recurrence equation and derivative rule formulas Assume a family of dierential equations is given for n N with continuous functions 0 x x and constants and we search for p olynomial solutions of degree n Then n since p x is linear one deduces that x must b e an at most linear p olynomial and since 1 p x is quadratic one deduces that x must b e an at most quadratic p olynomial Hence 2 we may assume that 2 x ax bx c x dx e n Equating co ecients of the highest p owers x in for generic p x given by one n deduces that moreover ann dn or ann dn n n Hence only if the dierential equation takes the sp ecial form 2 00 0 ax bx c y x dx e y x ann dn y x it can have p olynomial solutions Moreover we can assume that for n N hence an d for n N since n otherwise no orthogonal p olynomial solutions can exist This is discussed in detail in In particular d In the following theorem we give explicit representations of the corresp onding recurrence equation and
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