J. Math. Aal. Appl. 297 2004 186 193 www.elsevier.com/locate/jmaa Some families of geeratig fuctios for the multiple orthogoal polyomials associated with modified Bessel K-fuctios M.A. Özarsla, A. Altı Akara Uiversity, Faculty of Sciece, Departmet of Mathematics, Tadoğa 06100, Akara, Turkey Received 30 July 2003 Available olie 17 Jue 2004 Submitted by H.M. Srivastava Abstract The mai object of this paper is to derive several substatially more geeral families of biliear, bilateral, ad mixed multilateral fiite-series relatioships ad geeratig fuctios for the multiple orthogoal polyomials associated with the modified Bessel K-fuctios also kow as Macdoald fuctios. Some special cases of the above statemets are also give. 2004 Published by Elsevier Ic. Keywords: Macdoald fuctio; Multiple orthogoal polyomial; Rodrigues formula; Geeratig fuctio; Fiite-series relatioship 1. Itroductio We cosider multiple orthogoal polyomials associated with modified Bessel fuctio K ν, which were first itroduced by Va Assche ad Yakubovich [9] ad which were recetly also studied by Be Cheikh ad Douak [1]. The modified Bessel fuctio of the secod kid K ν x ν 0 is sometimes kow as the Macdoald fuctio, especially i the Russia literature. The scaled modified Bessel K-fuctio ρ ν is the defied as follows: * Correspodig author. E-mail addresses: ozarsla@sciece.akara.edu.tr M.A. Özarsla, alti@sciece.akara.edu.tr A. Altı. 0022-247X/$ see frot matter 2004 Published by Elsevier Ic. doi:10.1016/j.jmaa.2004.04.054
M.A. Özarsla, A. Altı / J. Math. Aal. Appl. 297 2004 186 193 187 ρ ν x = 2x ν/2 K ν 2 x, x>0. For the multiple orthogoal polyomials, we use the weight fuctios dµ 1 x = x α ρ ν x dx, dµ 2 x = x α ρ ν+1 x dx, x [0,, α > 1, ν 0. With these weight fuctios, we ca defie multiple orthogoal polyomials of types I ad II. Let, m ; the the vector A α,m x, Bα,m x of multiple orthogoal polyomials of type I is such that A α,m x is a polyomial of degree at most, Bα,m x is a polyomial of degree at most m ad A α,m x, Bα,m x satisfies the orthogoality coditios [ A α,m xρ ν x + B α,m xρ ν+1x ] x k+α dx = 0, k= 0, 1, 2,...,+ m. 0 We use the otatio q,m α x = Aα,m xρ νx + B,m α xρ ν+1x, Q α 2 x = qα, x, Qα 2+1 x = qα +1, x. The multiple orthogoal polyomials p,m α x of type II are such that pα,m x is a polyomial of degree at most + m that satisfies the multiple orthogoality coditio 0 0 p α,m xρ νxx k+α dx = 0, k= 0, 1, 2,..., 1, p α,m xρ ν+1xx k+α dx = 0, k= 0, 1, 2,...,m 1. I [9] it was show that the weights ρ ν,ρ ν+1 form a AT-system o [0, [6, p. 140] i fact, they are very close to a ikishi system or MT-system [6, p. 142], so that the polyomials A α,m x, Bα,m x, adpα,m x have degrees exactly, m ad + m, respectively. Usually, the polyomial p,m α x is chose to be moic. We defie P α 2 x = pα, x, P α 2+1 x = pα +1, x. These multiple orthogoal polyomials were itroduced i [9] ad solve a ope problem posed by Prudikov [7]. The various differetial properties of the modified Bessel K-fuctio imply useful differetial properties of the multiple orthogoal polyomials, a Rodrigues formula for type I polyomials, ad explicit formulas for the recurrece coefficiets i the four-term recurrece relatio of type II polyomials [9]. Recetly, type II polyomials were also studied i [1], who started with a explicit expressio of type II multiple orthogoal polyomials ad foud a geeratig fuctio ad a third-order differetial equatio for these polyomials. I [3] a explicit expressio for both types I ad II multiple orthogoal polyomials were obtaied.
188 M.A. Özarsla, A. Altı / J. Math. Aal. Appl. 297 2004 186 193 2. Geeratig fuctios for the multiple orthogoal polyomials of type I ad II I this sectio, let us remember some theorems i order to prove our mai theorems i the followig sectio. I [3] oe ca fid the followig theorem. Theorem A. We have the followig geeratig fuctio for x C \, 0]: Q α 1 xt! = 1 x α+1 ρ υ, Re x 0 x 0, t < 1 2. 1 If Re x<0ad Im x 0, the the series coverges for t < Imx x + Imx. The followig theorem is a very specialized case of [8, Eq. 2.619, p. 141] for r = 0, s = 2adt replaced by t. Theorem B. The geeratig fuctio P α Hx,t= x t α + 1 α + ν + 1! is give by Hx,t= e t 0F 2 ; α + 1,α+ ν + 1; xt, x,t C. 2 3. Mai theorems I recet years by makig use of the familiar group-theoretic Lie algebraic method a certai mixed trilateral fiite-series relatioships have bee proved for orthogoal polyomials see, for istace, [8]. The mai object of this sectio is to derive several substatially more geeral families of biliear, bilateral, mixed multilateral fiite-series relatioship ad geeratig fuctios for the multiple orthogoal polyomials of type I ad II associated with the modified Bessel K-fuctios as Theorems 3.1 ad 3.2, respectively. By applyig the formula 1 we ca prove the followig theorem for multiple orthogoal polyomials of type I associated with the modified Bessel K-fuctio of the first kid, istead of usig group theoretic method, with the help of the similar method as cosidered i [2,5,10]. Theorem 3.1. Correspodig to a idetically ovaishig fuctio Ω µ ξ 1,ξ 2,...,ξ s of s real or complex variables ξ 1,ξ 2,...,ξ s s := 0 \{0} ad of complex order µ, let Λ 1 ψ,µ ξ 1,...,ξ s ; τ:= a k Ω µ+ψk ξ 1,...,ξ s τ k a k 0, ψ C. 3
M.A. Özarsla, A. Altı / J. Math. Aal. Appl. 297 2004 186 193 189 Suppose also that Θ,q µ,λ,ψ! x; ξ 1,...,ξ s ; ζ:= qk! a kq α+λk qk 1 xω µ+ψkξ 1,...,ξ s ζ k, q ; α + λk > 1. 4 The Θ,q µ,λ,ψ x; ξ 1,...,ξ s ; η = 1 x α+1 ρ ν t q t! Λ 1 ψ,µ ξ 1,...,ξ s ; η λ, 5 provided that each member of 5 exists ad that Re x 0 x 0, t < 1/2. IfRe x<0 ad Im x 0, the the series coverges for t < Imx x + Imx. The otatio meas the greatest iteger less tha or equal to /q. Proof. For coveiece, let x, t deote the first member of the assertio 5. The, upo substitutig for the polyomials Θ,q µ,λ,ψ x; ξ 1,...,ξ s ; ζ from the defiitio 4 ito the left-had side of 5, we obtai x, t = = 1 qk! a kq α+λk qk 1 xω µ+ψkξ 1,...,ξ s η k t qk a k Ω µ+ψk ξ 1,...,ξ s η k Q α+λk 1 xt!, which i view of 1 with α α + λk k 0 yields x, t = a k Ω µ+ψk ξ 1,...,ξ s η k 1 x λk+α+1 ρ ν { = a k Ω µ+ψk ξ 1,...,ξ s η λ } k 1 x α+1 ρ ν ad the assertio 5 follows immediately by meas of the defiitio 3. I a similar maer, by appealig to the formula 2, we are led fairly easily to
190 M.A. Özarsla, A. Altı / J. Math. Aal. Appl. 297 2004 186 193 Theorem 3.2. Correspodig to a idetically ovaishig fuctio Ω µ ξ 1,ξ 2,...,ξ s of s real or complex variables ξ 1,ξ 2,...,ξ s s := 0 \{0} ad of complex order µ, let Λ 2 ψ,µ ξ 1,...,ξ s ; τ:= a k Ω µ+ψk ξ 1,...,ξ s τ k a k 0, ψ C. 6 Suppose also that Φ,q µ,ψ x; ξ! 1,...,ξ s ; ζ:= qk! a kp qk α x Ω µ+ψk ξ 1,...,ξ s ζ k α + 1 qk α + ν + 1 qk, q, x C. 7 The, for t C we have Φ,q µ,ψ x; ξ 1,...,ξ s ; η t t q! = e t 0F 2 ; α + 1,α+ ν + 1; xtλ 2 ψ,µ ξ 1,...,ξ s ; η, 8 provided that each member of 8 exists. 4. Some special cases of Theorems 3.1 ad 3.2 Whe the multivariable fuctio Ω µ+ψk ξ 1,...,ξ s k 0,s is expressed i terms of simpler fuctio of oe ad more variables the we ca give further applicatios of Theorems 3.1 ad 3.2. For example, if we set s = 1, ξ 1 = z ad Ω µ+ψk z = y m z; µ + ψk,β k,m 0, µ, ψ C i Theorem 3.1, where y m z; α, β deotes the geeralized Bessel polyomials defied by see [4] y m z; α, β = 2 F 0 m, α + m 1; ; x β we shall readily obtai a class of bilateral geeratig fuctios for the geeralized Bessel polyomials or the multiple orthogoal polyomials of type I associated with modified Bessel K-fuctios, give by Corollary 4.1. If ad Π 1 µ,ψ,m z; τ:= a k y m z; µ + ψk,βτ k a k 0, ψ,µ,β C Ψ 1 µ,ψ,λ,,q x,z,ζ:=! qk! a kq α+λk qk 1 xy mz; µ + ψk,βζ k, q, λ C,,
M.A. Özarsla, A. Altı / J. Math. Aal. Appl. 297 2004 186 193 191 the Ψ 1 µ,ψ,λ,,q x,z, η t t q! = 1 x α+1 ρ ν Π 1 µ,ψ,m z; η λ, 9 provided that each member of 9 exists, where Re x 0 x 0, t < 1/2. IfRe x<0 ad Im x 0, the the series coverges for t < Imx x + Imx. Example 4.1. By usig the geeratig relatio µ + m + k 2 y k m z; µ + k,βτ k = 1 τ 1 µ m y m z 1 τ ; µ, β, τ < 1, for geeralized Bessel polyomials see [5, Eq. 4.15, p. 270] ad takig µ + m + k 2 ψ = 1, a k =, k we have 1 µ + m + k 2 Q α+λk qk! k qk 1 xy mz; µ + k,βη k t qk 1 x = α+1 ρ η 1 µ m z λ ν 1 λ y m λ η ; µ, β, η λ < 1. We ca give aother special case of Theorem 3.1 whe we set s = 1, ξ 1 = z ad Ω µ+ψk z = Q µ+ψk z k, 0,µ+ ψk > 1 we shall readily obtai a class of biliear geeratig fuctio for the multiple orthogoal polyomials of type I associated with modified Bessel K-fuctios, give by Corollary 4.2. If ad Π 2 µ,ψ, z; τ:= a k Q µ+ψk zτ k a k 0 [/q Ψ 2 µ,ψ,λ,,q x,z,ζ:=! qk! a kq α+λk, q, λ R, qk 1 xqµ+ψk z ζ k
192 M.A. Özarsla, A. Altı / J. Math. Aal. Appl. 297 2004 186 193 the Ψ 2 µ,ψ,λ,,q x,z, η t t q! = 1 x α+1 ρ ν Π 2 η µ,ψ, z; λ, 10 provided that each member of 10 exists, where Re x 0 x 0, t < 1/2. IfRe x<0 ad Im x 0, the the series coverges for t < Imx x + Imx. By choosig s = 1, ξ 1 = z ad Ω µ+ψk z = Q µ+ψk z where µ + ψk > 1 i Theorem 3.2 we have the followig result immediately. Corollary 4.3. If ad Π 3 µ,ψ, z; τ:= a k Q µ+ψk zτ k a k 0 Ψ 3 µ,ψ,λ,,q x,z, η! t q = qk! a kp qk α x Q µ+ψk z ζ k α + 1 qk α + ν + 1 qk, q, the Ψ 3 µ,ψ,λ,,q x,z, η t t q! = e t 0F 2 ; α + 1,α+ ν + 1; xtπ 3 µ,ψ, z; τ, 11 provided that each member of 11 exists. Moreover, for each suitable choice of the coefficiets a k k 0, if the multivariable fuctio Ω µ+ψk ξ 1,...,ξ s s is expressed as a appropriate product of several simpler fuctios as it see i the example, Theorems 3.1 ad 3.2 ca be show to yield various classes of mixed multilateral geeratig fuctios for the multiple orthogoal polyomials of type I ad II associated with modified Bessel K-fuctios. Ackowledgmet We thak referee for his/her valuable suggestios that improved the presetatio of the paper.
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