ON A CLASS OF ORTHOGONAL POLYNOMIALS

Ioan Gavrea

ON A CLASS OF ORTHOGONAL POLYNOMIALS

Ioan Gavrea

2007

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In this note we study a system of polynomials {b Pk} orthogonal with respect to the modified measure db (t) = t d t c w(t)dt, t 2 (0,1) where d,c < 0 and w is a weight function, using orthogonal polynomials {Pk} with respect to the measure dw(t).

STUDIA UNIV. “BABEŞ–BOLYAI”, MATHEMATICA, Volume LII, Number 3, September 2007 ON A CLASS OF ORTHOGONAL POLYNOMIALS I. GAVREA Dedicated to Professor Petru Blaga at his 60th anniversary Abstract. In this note we study a system of polynomials {Pbk } orthogonal with respect to the modified measure b = dλ(t) t−d w(t)dt, t−c t ∈ [0, 1] where d, c < 0 and w is a weight function, using orthogonal polynomials {Pk } with respect to the measure dw(t). 1. Introduction In [3] G.V. Milovanovic, A.S. Cvetković and M.M. Matejic investigated polynomials orthogonal with respect to the moment functional 1 1 t+ c+ p 2 c 1 − t2 dt, L(P ) = P (t) 1 1 −1 t+ c+ 2 2c Z 1 P ∈Π where c ∈ R \ {0}. Similar measures, e.g. with the weight function (1 − t2 )(1 − k 2 t2 )−1/2 , k 2 < 1 were studied in [4]. For the Chebyshev measure of the first kind the same modification has been studied in [2]. In this note we investigate polynomials orthogonal with respect to the moment functional A(P ) = Z 1 0 P (t) t−d w(t)dt, t−c P ∈Π Received by the editors: 08.04.2007. 2000 Mathematics Subject Classification. 33C45, 33C47. Key words and phrases. orthogonal polynomials, weight function, Chebyshev measure. 95 (1.1) I. GAVREA where dw(t) is a positive measure on [0, 1] having finite moments of all orders, d, c being fixed negative numbers. We will denote by {Pk } the set of orthogonal polynomials with respect to the measure dw(t) and by {Pbk } the system of orthogonal polynomials with respect to the measure dw1 (t) = t−d w(t)dt. t−c The existence of {Pbk } is guaranteed, since dw1 (t) is a positive measure on [0, 1] having finite moments of all orders Z 1 t−d w(t)dt. A(tk ) = tk t−c 0 We will solve the problem in two steps as well as in [3]. First, we consider modification of dw(t) measure by linear factor t − d computing the coefficients of the three-term recurrence relation and then we consider the modification of dw(t) measure by the linear divisor. In the following we will suppose that Z 1 w(t)dt = 1. (1.2) 0 2. Linear factors We denote by w e the weight function defined by w(t) e = (t − d)w(t). (2.1) It is well known (see [5]) that the orthogonal polynomial of degree n relative to the weight function w(t) e is given by Pen (t) = n X Pk (t)Pk (d). (2.2) k=0 It is known ([5]) that there exists a relation of the form Pen (t) = (An t + Bn )Pen−1 (t) − cn Pen−2 (t), n = 2, 3, . . . (2.3) Theorem 2.1. The coefficients An , Bn , Cn are given by: An = 96 Pn,n (d) Pn−1,n−1 (d) (2.4) ON A CLASS OF ORTHOGONAL POLYNOMIALS Bn = 1 + 2 Pn,n (d) an−1,n−1 dPn,n (d) · 2 − an,n Pn−1,n−1 (d) Pn−1,n−1 (d) (2.5) 2 Pn,n (d) an−1,n−1 · 2 an,n Pn−1,n−1 (d) Cn = (2.6) where Pn,n is the polynomial of degree n orthogonal with respect to w(t) normalized by Z 1 xn Pn,n (x)w(x)dx = 1 0 and an,n , n = 0, 1, . . . is the coefficient of xn of the polynomial Pn,n . Proof. For k ∈ {0, 1, . . . , n} we denote by Pn,k the polynomial of degree n defined by the equalities: Z 1 w(t)ti Pn,k (t) = δk,i , i = 0, n. (2.7) 0 Pn,k is well defined by the relations (2.7). If P is a polynomial of degree n then P can be written in the following form: P (t) = n X Mk Pn,k (t) (2.8) k=0 where Mk = Z 1 tk P (t)w(t)dt. 0 The polynomial Pen can be written as Pen (t) = We note that n X dk Pn,k (t) = tk Pn,k (d). (2.9) k=0 k=0 Z n X 1 Pn (t)w(t)dt = 1. (2.10) 0 From (2.9) and (2.10) the equality (2.3) is equivalent with n X k=0 dk Pn,k (t) = (An t + Bn ) n−1 X dk Pn−1,k (t) − Cn k=0 n−1 X dk Pn−2,k (t), n ≥ 2. (2.11) k=0 From (2.10) and (2.11) we get: 1 = Bn − Cn + dAn . (2.12) 97 I. GAVREA On the other hand (2.11) can be written as: n X tk Pn,k (d) = (An t + Bn ) n−1 X tk Pn−1,k (d) − Cn k=0 k=0 n−2 X tk Pn−2,k (d). (2.13) k=0 From (2.13) we get An = Pn,n (d) Pn−1,n−1 (d) (2.14) and Pn,n−1 (d) = An Pn−1,n−2 (d) + Bn Pn−1,n−1 (d). (2.15) Pn,n−1 can be written as: Pn,n−1 (x) = Pn−1,n−1 (x) + aPn,n (x) where a=− Z (2.16) 1 tn Pn−1,n−1 (t)dt. 0 By (2.15), (2.14) and (2.16) we obtain Bn = + Z 2 Pn−1,n−1 (d) − Pn,n (d)Pn−2,n−2 (d) 2 Pn−1,n−1 (d) 1 tn−1 (Pn−2,n−2 (t) − tPn−1,n−1 (d))w(t)dt · Pn−1,n−1 (d)Pn,n (d) 0 2 Pn−1,n−1 (d) . (2.17) There is the constants αn , βn , γn such that Pn,n (t) = (αn t + βn )Pn−1,n−1 (t) − γn Pn−2,n−2 (t) (2.18) and we have αn = γn = and βn = an,n an−1,n−1 Z an,n an−1,n−1 (2.19) 1 tn−1 (Pn−2,n−2 (t) − tPn−1,n−1 (t))w(t)dt. (2.20) 0 From (2.17), (2.18), (2.19) and (2.20) we get Bn = 1 + 98 2 Pn,n (d) an−1,n−1 Pn,n (d) · 2 −d . an,n Pn−1,n−1 (d) Pn−1,n−1 (d) (2.21) ON A CLASS OF ORTHOGONAL POLYNOMIALS By (2.12) and (2.21) we obtain 2 (d) Pn,n an−1,n−1 · 2 an,n Pn−1,n−1 (d) Cn = and the theorem is proved. 3. Linear divisors Let us consider the weight function w2 (t) = w(t) e . t−c The orthogonal polynomials {Pbn }, n = 0, 1, . . . are orthogonal polynomials relative to the weight function w2 (t). Theorem 3.1. The polynomial Pbn is given by Z 1 e Pn−1,n−1 (t) w(t)dt t−c Pbn (t) = Pen−1,n−1 (t) − Pen,n (t) 0 Z 1 Pen,n (t) w(t)dt t−c 0 (3.1) where Pen,n is orthogonal polynomial of degree n relative to the weight function w e normalized by Z 1 0 Pbn is normalized by Z 0 xn Pen,n (x)w(x)dx e = 1. 1 xn Pbn (x)w(x)dx = 1. Proof. By the conditions Z 1 xk Pbn (x)w2 (x) = 0 for k = 0, 1, . . . , n − 1 0 we get Z Z 1 k x − ck e xk b Pn (x)w(x)dx e = Pn (x)w(x)dx e x−c 0 x−c 0 Z 1 k Z 1 e x − ck e Pn (x)w(x) e dx = Pn (x)w(x)dx e +ck x−c x−c 0 0 0= 1 99 I. GAVREA or 0= k−1 X i c Z 1 0 i=0 If we denote by tk−i−1 Pbn (t)w(t)dt, e Mi = Z k = 1, 2, . . . , n − 1. (3.2) 1 0 from (3.2) we get ti w(t) e Pbn (t)dt M0 = M1 = · · · = Mn−2 = 0 and so, the polynomial Pbn can be written Pbn (t) = Mn−1 Pen,n−1 (t) + Mn Pen,n (t). By the condition Z t Pbn (t)w(t) e dt = 1 t−c 1 n 0 we get Mn−1 = 1. From the condition Z 1 0 we obtain Pbn (t) w(t)dt e =0 t−c Z Pen,n−1 (t)w(t) e dt t−c 0 Mn = − Z . 1 Pn,n (t)w(t) e dt t−c 0 1 The polynomial Pen,n−1 can be written as where Pen,n−1 (t) = Pen−1,n−1 (t) + un Pen,n (t) un = − Z 0 (3.4) (3.5) 1 tn Pen−1,n−1 (t)w(t)dt. e From (3.6), (3.5), (3.4) and (3.3) we get Z 1 e Pn−1,n−1 (t) w(t)dt e t−c 0 b e e Pn (t) = Pn−1,n−1 (t) − Pn,n (t) Z 1 . Pen,n (t) w(t)dt e t−c 0 The last relation proves the theorem. 100 (3.3) (3.6) ON A CLASS OF ORTHOGONAL POLYNOMIALS References [1] Gautschi, W., Orthogonal Polynomials: Computation and Approximation, Oxford University Press Inc., New York, 2004. [2] Grinšpun, Z.S., On a class of orthogonal polynomials, (Russian), Vestnik Leningrad Univ. 21(1966), 147-149. [3] Milovanović, G.V., Cvetkovič, A.S., Matejič, M.M., Remark on orthogonal polynomials induced by the modified Chebyshev measure of the second kind, Facta Universitatis (Niš), Ser. Math. Inform., 21(2006), 13-21. [4] Rees, C.J., Eliptic orthogonal polynomials, Duke Math. J., 12(1945), 173-187. [5] Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 23, 4th ed., Amer. Math. Soc. Providence, R.I., 1975. Technical University of Cluj-Napoca, Department of Mathematics Daicoviciu 15, Cluj-Napoca, Romania E-mail address: ioan.gavrea@math.utcluj.ro 101

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