Generating functions on extended jacobi polynomials from lie group view point

Publicacions Matem`atiques, Vol 40 (1996), 3–13. GENERATING FUNCTIONS ON EXTENDED JACOBI POLYNOMIALS FROM LIE GROUP VIEW POINT Manik Chandra Mukherjee Abstract Generating functions play a large role in the study of special functions. The present paper deals with the derivation of some novel generating functions of extended Jacobi polynomials by the application of group-theoretic method introduced by Louis Weisner. In fact, by suitably interpreting the index (n) and the parameter (β) of the polynomial under consideration we define four linear partial differential operators and on showing that they generate a Lie-algebra, we obtain a new generating relation (3.3) as the main result of our investigation. Furthermore, some generating functions of Laguerre, Hermite, Bessel and Jacobi polynomials are obtained as the special cases of our main result. Some applications of our results are also pointed out. 1. Introduction The extended Jacobi polynomials as defined by I. Fujiwara [1] are as follows: (1.1) Fn(α, β;x)=(−1)n n!λ b−an (x−a)−α(b−x)−β ×Dn[(x−a)n+α(b−x)n+β],where D≡d dx. They satisfy the following ordinary differential equation: (1.2) [(x−a)(b−x)D2+{(α+ 1)(b−x)−(β+ 1)(x−a)}D +n(1+α+β+n)]y=0. Recently, some attempts [2], [3] have been made by researchers for deriving generating functions of the polynomials under consideration from 4M. C. Mukherjee the Lie group view point. The aim at presenting this article is to apply L. Weisner’s group-theoretic method [4] with the simultaneous suitable interpretations of the index (n) and the parameter (β) in the study of extended Jacobi polynomials. It may be mentioned that in the course of constructing a four dimensional Lie algebra we have obtained two operators such that, when operated on the polynomials under consideration, simultaneously raise (lower) and lower (raise) the index and the parameter by one unit. Such type of operators do not seem to have appeared earlier in the study of extended Jacobi polynomials. The main results of this paper are the formulas (3.3) to (3.6) given in Section 3. 2. Group Theoretic Method Replacing d dx by ∂ ∂x,nby y∂ ∂y ,βby z∂ ∂z and uby v(x, y, z) in (1.2), we get the following partial differential equation: (2.1) (x−a)(b−x)∂2v ∂x2+(a−x)z∂2v ∂x∂z +yz ∂2v ∂y∂z +y2∂2v ∂y2+{(1+α)(b−x)−(x−a)}∂v ∂x +(2+α)y∂v ∂y =0. Thus v1(x, y, z)=Fn(α, β;x)ynzβis a solution of (2.1) since Fn(α, β;x) is a solution of (1.2). Let us now introduce a set of linear partial differential operators, Ai,i=1,2,3,4, defined as follows: (2.2)                                  A1=y∂ ∂y, A2=z∂ ∂z, A3=(x−a)y−1z λ ∂ ∂x −z λ ∂ ∂y, A4=λ b−a(x−a)(x−b)yz−1∂ ∂x +(x−b)y2z−1∂ ∂y +(x−a)y∂ ∂z +(1+α)(x−b)yz−1. Then (2.3) A1(Fn(α, β;x)ynzβ)=nFn(α, β;x)ynzβ, A2(Fn(α, β;x)ynzβ)=βFn(α, β;x)ynzβ, A3(Fn(α, β;x)ynzβ)=(n+α)Fn−1(α, β +1;x)yn−1zβ+1, A4(Fn(α, β;x)ynzβ)=(n+1)Fn+1(α, β −1; x)yn+1zβ+1. On generating functions of extended Jacobi polynomials 5 We now proceed to find the commutator relations satisfied by Ai (i=1,2,3,4). Using the notation [A, B]u=(AB −BA)u we get (2.4) [A1,A 2]=0, [A1,A 3]=−A3, [A1,A 4]=A4, [A2,A 3]=A3, [A2,A 4]=−A4and [A3,A 4]=2A1+(1+α). So from the above commutator relations we can easily state the following: Theorem. The set of operators {1,A i(i=1,2,3,4)}where 1 stands for the identity operator, generates a Lie-algebra L. Now the partial differential operator Lgiven by L=(x−a)(b−x)∂2 ∂x2+(a−x)z∂2 ∂x∂z +yz ∂2 ∂y∂z +y2∂2 ∂y2+{(1+α)(b−x)−(x−a)}∂ ∂x +(2+α)∂ ∂y can be expressed as follows: (2.5) (x−a)L=(b−a)(A4A3+A2 1+αA1). We can easily verify that each of Ai(i=1,2,3,4) commutes with L.In other words, (2.6) [(x−a)L, Ai]=0. Now the extended forms of the groups generated by Ai(=1,2,3,4) are given as follows: (2.7) ea1A1f(x, y, z)=f(x, ea1y,z), (2.8) ea2A2f(x, y, z)=f(x, y, ea2z), (2.9) ea3A3f(x, y, z)=fay−a3z λ+(x−a)y y−a3z λ ,y−a3 z λ,z , (2.10) ea4A4f(x, y, z)=1−λ b−a(x−b)a4 y z−(1+α) ×f  x−aλ b−a(x−b)a4y z 1−λ b−a(x−b)a4y z ,y 1−λ b−a(x−b)a4y z ,z1+λa4y z 1−λ b−a(x−b)a4y z . 6M. C. Mukherjee Thus we have ea4A4ea3A3ea2A2ea1A1f(x, y, z)=1−λ b−a(x−b)a4 y z−(1+α) (2.11) ×f  λy x−aλ b−a(x−b)a4y z−aa3z1+λa4y z1−λ b−a(x−b)a4y z 1−λ b−a(x−b)a4y zλy−a3z1+λa4y z , ea1  λy −a3z1+λa4y z λ1−λ b−a(x−b)a4y z ,ea2z1+λa4y z 1−λ b−a(x−b)a4y z . 3. Generating Functions From (2.1) it is seen that Fn(α, β;x)ynzβis a solution of the system: (A1−n)u=0 Lu =0 ;(A2−β)u=0 Lu =0 ;(A1+A2−n−β)u=0 Lu =0 . From (2.5) we observe that S(x−a)L(Fn(α, β;x)ynzβ)=(x−a)LS(Fn(α, β;x)ynzβ)=0, where S=ea4A4ea3A3ea2A2ea1A1. Therefore, the transformation S(Fn(α, β;x)ynzβ) is annihilated by (x−a)L. Now putting α1=α2= 0 and replacing f(x, y, z)by Fn(α, β;x)ynzβwe get ea4A4ea3A3(Fn(α, β;x)ynzβ) (3.1) =1−λ b−a(x−b)a4 y z−(1+α+β+n)1+λa4 y zβ ×1−a3z λy 1+λa4 y zn Fn  α,β, λyx−aλ b−a(x−b)a4y z−aa3z1+λa4y z1−λ b−a(x−b)a4y z 1−λ b−a(x−b)a4y zλy−a3z1+λa4y z  . On generating functions of extended Jacobi polynomials 7 But ea4A4ea3A3(Fn(α, β;x)ynzβ)(3.2) =∞  k=0 n+k  p=0 (a3)p p! (a4)k k!(−1)p(−n−α)p(n−p+1) k ×Fn−p+k(α, β +p−k;x)yn−p+kzβ+p−k. Equating (3.1) and (3.2) we get our main result: 1−λ b−e(x−b)a4 y z−(1+α+β+n)1+λa4 y zβ (3.3) ×1−a3z λy 1+λa4 y zn Fn α,β, λyx−aλ b−a(x−b)a4y z−aa3z1+λa4y z1−λ b−a(x−b)a4y z 1−λ b−a(x−b)a4y zλy−a3z1+λa4y z   =∞  k=0 n+k  p=0 (a3)p p! (a4)k k!(−1)p(−n−α)p(n−p+1) k ×Fn−p+k(α, β +p−k;x)yn−p+kzβ+p−k. Before discussing the particular cases of (3.3) we would like to point out that the operators A3,A4being non commutative, the relation (3.3) will change if we change the order of the Lie element ea4A4ea3A3. This is done in Section 4. Now we discuss several cases: Case 1. Putting a4=0,a3= 1 and −z y=tin (3.3) we get (3.4) 1+ t λn Fnα, β;x+at λ 1+ t λ =∞  p=0 (−n−α)p p!Fn−p(α, β +p;x)tp. 8M. C. Mukherjee Case 2. Putting a3=0,a4= 1 and y z=tin (3.3) we get (3.5) 1−λ b−a(x−b)t−(1+α+β+n) (1+λt)β×Fnα, β, x−aλ b−a(x−b)t 1−λ b−a(x−b)t =∞  k=0 (n+1) k k!Fn+k(α, β −k;x)tk. Case 3. Putting a3=1,a4=1 wand y z=tin (3.3) we get 1−λ b−a(x−b)t w−(1+α+β+n) 1+λt wβ1−1 λt 1+λt wn (3.6) ×Fn α, β, λy x−aλ b−a(x−b)t w−az 1+λt w1−λ b−a(x−b)t w 1−λ b−a(x−b)t wλy−z1+ t w   =∞  k=0 n+k  p=0 1 wk 1 k! (−1)p(−n−α)p p!(n−p+1) k ×Fn−p+k(α, β +p−k;x)tk−p. We now proceed to find some particular cases of interest of results (3.4) and (3.5). Particular Case 1: (On Jacobi Polynomials). Putting −a=b= 1 and λ= 1 in (3.4) and (3.5) we get the following generating relations of Jacobi polynomials [8] (3.7) (1 + t)np(β,α) nx−t 1+t=∞  p=0 (−n−α)p p!p(β+p,α) n−p(x)tp and (3.8) 1−t 2(x−1)−(1+α+β+n) (1+t)βp(β,α) nx+t 2(x−1) 1−t 2(x−1)  =∞  k=0 (n+1) k k!p(β−k,α) n+k(x)tk. On generating functions of extended Jacobi polynomials 9 Now by using the symmetry relation p(α,β) n(−x)=(−1)np(β,α) n(x) we get the following generating relation [9] (3.9) (1 + t)np(α,β) nx−t 1+t=∞  p=0 (−n−α)p p!p(α+p,β) n−p(x)tp and (3.10) (1 + t)α1−t 2(x−1)−(1+α+β+n) p(α,β) nx−t 2(x−1) 1−t 2(x−1)  =∞  k=0 (n+1) k k!p(α−k,β) n+k(x)tk. Particular Case 2: (On Laguerre Polynomials). Putting a=0,λ=1,β=band taking limit as β→∞, we get the following generating relations of Laguerre polynomials [6] (3.11) (1 −t)−1−α−nexp −tx 1−tL(α) nx 1−t =∞  k=0 (n+1) k k!L(α) n+k(x)tk and (3.12) (1 −t)nL(α) nx 1−t=∞  p=0 (−n−α)p p!L(α) n−p(x)tp. Particular Case 3: (On Hermite Polynomials). Putting α=β,−a=b=√a,λ=2 √αand taking limit as α→∞,we get the following generating relations of Hermite polynomials [5] (3.13) Hn(x+y)= n  p=0 n pHn−p(x)(2y)p and (3.14) e2xt−t2Hn(x−t)= ∞  k=0 1 k!Hn+k(x)tk. 10 M. C. Mukherjee Particular Case 4: (On Bessel Polynomials). Replacing xby 1+ 2xε s,tby sw ε, putting −a=b=λ=1,α=v−ε−1, β=ε−1 and then taking the limit as ε→∞, we get the following generating relations of Bessel’s polynomials [7] (3.15) esw(1 −xw)1−ν−nYnx 1−xw ;ν,s  =∞  k=0 ekYn+k(x;ν−k,s)wk k! and (3.16) (1 + t)nYnw 1+t;ν,s = n  p=0 n pYn−p(x;ν+p, s)tp. 4. Variants of the Result (3.3) Since [A3,A 4]= 0, we can well aply the operator ea3A3×ea4A4on the function Fn(α, β;x)ynzβ. Now ea3A3ea4A4(Fn(α, β;x)ynzβ) (4.1) =1−a4a3+λ b−a(x−b)y z−(1+α+β+n)1−a3z λy n ×1+a4 z(λy −a3z)β ×Fn xy−aa3z λ−aa4y−a3z λa3+λ b−a(x−b)y z y−a3z λ1−a4a3+λ b−a(x−b)y z  . Again ea3A3ea4A4(Fn(α, β;x)ynzβ)(4.2) =∞  k=0 n+k  p=0 (a3)p p! (a4)k k!(−1)p(−α−n−k)p(n+1) k ×Fn+k−p(α, β −k+p;x)y zk−p . On generating functions of extended Jacobi polynomials 11 Equating (4.1) and (4.2) we get the following result 1−a4a3+λ b−a(x−b)y z−(1+α+β+n)1−a3z λy n (4.3) ×1+a4 z(λy −a3z)β ×Fn xy −aa3z λ−aa4y−a3z λa3+λ b−a(x−b)y z y−a3z λ1−a4a3+λ b−a(x−b)y z   =∞  k=0 n+k  p=0 (a3)p p! (a4)k k!(−1)p(−α−n−k)p(n+1) k ×Fn+k−p(α, β −k+p;x)y zk−p . Application Relations (3.4) and (3.5) may be applied in deriving bilateral generating functions involving the special function under consideration. We shall give an application by using the relation (3.5) in deriving the following theorem as bilateral generating relation. Theorem. If (i) G(x, w)= ∞  n=0 anFn(α, β;x)wn then (1+λt)β1−λ b−a(x−b)t−(1+α+β) (ii) ×Gx−λa b−a(x−b)t 1−λ b−a(x−b)t,zt 1−λ b−a(x−b)t =∞  n=0 tnσn(x, z) where σn(x, z)= ∞  k=0 ak (k+1) n−k (n−k)! Fn(α, β −n+k;x)zk.