J. Korean Math. Soc. 46 (2009), No. 6, pp. 1277–1292 DOI 10.4134/JKMS.2009.46.6.1277
GENERATING RELATIONS INVOLVING 3-VARIABLE 2-PARAMETER TRICOMI FUNCTIONS USING LIE-ALGEBRAIC TECHNIQUES Subuhi Khan, Mumtaz Ahmad Khan, and Rehana Khan Abstract. This paper is an attempt to stress the usefulness of the multivariable special functions. In this paper, we derive generating relations involving 3-variable 2-parameter Tricomi functions by using Lie-algebraic techniques. Further we derive certain new and known generating relations involving other forms of Tricomi and Bessel functions as applications.
1. Introduction The function (1.1)
Cn (x) =
∞ X (−1)r xr , r! (n + r)! r=0
is a Bessel like function known as Tricomi function and is characterized by the following link with the ordinary Bessel function Jn (x) [11]: √ (1.2) Cn (x) = x−n/2 Jn 2 x or (1.3)
Jn (x) =
x n 2
Cn
x2 4
.
The Tricomi function Cn (x) satisfies the generating function ∞ X x (1.4) exp t − = Cn (x) tn , t n=−∞ Received April 21, 2008. 2000 Mathematics Subject Classification. 33C10, 33C80, 33E20. Key words and phrases. generalized Tricomi functions, Lie-algebra representation, generating relations. This work has been done under a Major Research Project No. F.33-110/2007 (SR) sanctioned to the first author by the University Grants Commission, Government of India, New Delhi. c °2009 The Korean Mathematical Society
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which yield the recurrences (1.5)
d Cn (x) = −Cn+1 (x), dx x Cn+1 (x) − nCn (x) + Cn−1 (x) = 0.
On combining the above recurrence relations, we get the following differential equation satisfied by Cn (x): d2 d + 1 Cn (x) = 0. (1.6) x 2 + (n + 1) dx dx Equation (1.6) ensures that Cn (x) are eigen functions of the operator d d d x −n . dx dx dx The above operator, whose importance has been recognized within the frame work of theory of monomiality of Laguerre polynomials [2], can be viewed as a generalization of the ordinary derivative so that Cn (x) can be considered as a generalization of the ordinary exponential function, it does not possess the semigroup property and indeed Cm (x + y) 6= Cm (x)Cm (y). This fact far from being a limitation, allows the possibility of introducing other families of Bessel like functions.
(1.7)
On (x) = −
The study of the properties of multi-variable generalized special functions has provided new means of analysis for the solutions of large classes of partial differential equations often encountered in physical problems. The relevance of the special functions in physics is well established. Most of the special functions of mathematical physics as well as their generalizations have been suggested by physical problems. In order to further stress the usefulness of the generalized special functions, Dattoli et al. [3] have introduced the three variable two parameter extension of Tricomi functions defined as: ∞ X (1.8) Cn (x, y, z; τ1 , τ2 ) = τ2l Cn−3l (x, y; τ1 ) Cl (z). l=−∞
The generating function for 3-variable 2-parameter Tricomi functions (3V2PTF) Cn (x, y, z; τ1 , τ2 ) is given as: ∞ X x y z 2 3 (1.9) exp t − + t τ1 − 2 + t τ2 − 3 = tn Cn (x, y, z; τ1 , τ2 ). t t τ1 t τ2 n=−∞ The 3V2PTF Cn (x, y, z; τ1 , τ2 ) are related to the 3-variable 2-parameter Bessel functions (3V2PBF) Jn (x, y, z; τ1 , τ2 ) by [3] r ! √ x x3 √ √ −n/2 (1.10) Cn (x, y, z; τ1 , τ2 ) = x Jn 2 x, 2 y, 2 z; √ τ1 , τ2 , y z
TRICOMI FUNCTIONS
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or (1.11)
Jn (x, y, z; τ1 , τ2 ) =
x n 2
Cn
x2 y 2 z 2 2y 4z , , ; 2 τ1 , 3 τ2 , 4 4 4 x x
where Jn (x, y, z; τ1 , τ2 ) is given by [3] (1.12)
Jn (x, y, z; τ1 , τ2 ) =
∞ X
τ2l Jn−3l (x, y; τ1 ) Jl (z).
l=−∞
with the generating function ∞ X
(1.13)
Jn (x, y, z; τ1 , τ2 )tn
n=−∞
x 1 y 2 1 z 3 1 = exp t− + t τ1 − 2 + t τ2 − 3 . 2 t 2 t τ1 2 t τ2
The theory of special functions from the group-theoretic point of view is a well established topic, providing a unifying formalism to deal with the immense aggregate of the special functions and a collection of formulae such as the relevant differential equations, integral representations, recurrence formulae, composition theorems, etc., see for example [13, 14]. The first significant advance in the direction of obtaining generating relations by Lie-theoretic method is made by Weisner [15-17] and Miller [10]. Within the group-theoretic context, indeed a given class of special functions appears as a set of matrix elements of irreducible representation of a given Lie group. The algebraic properties of the group are then reflected in the functional and differential equations satisfied by a given family of special functions, whilst the geometry of the homogeneous space determines the nature of the integral representation associated with the family. Recently some contributions related to Lie-theoretical representations of generalized Laguerre and Hermite polynomials and Bessel functions of two variables have been given, see for example Khan [7], Khan and Pathan [8] and Khan et al. [9]. The 3-dimensional complex local Lie group T3 is the set of all 4 × 4 matrices of the form 1 0 0 τ 0 e−τ 0 c , b, c, τ ∈ C. (1.14) g = 0 0 eτ b 0 0 0 1
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A basis for the Lie algebra T3 = L(T3 ) (1.15) 0 0 0 0 0 0 0 0 0 0 − 0 0 J+ = 0 0 0 1 , J = 0 0 0 0 0 0 0 0
is provided by the matrices 0 0 0 0
0 0 1 , J 3 = 0 0 0 0 0
0 0 1 −1 0 0 , 0 1 0 0 0 0
with commutation relations [J 3 , J + ] = J + , [J 3 , J − ] = −J − , [J + , J − ] = 0.
(1.16)
Also, the Lie algebra E3 of the Euclidean group E3 (real 3-parameter global Lie group) has basis elements 0 0 0 0 −1 0 0 0 1 0 0 , (1.17) J1 = 0 0 0 , J2 = 0 0 1 , J3 = 1 0 0 0 0 0 0 0 0 0 with commutation relations (1.18)
[J1 , J2 ] = 0, [J3 , J1 ] = J2 , [J3 , J2 ] = −J1 .
Further, we observe that the complex matrices (1.19)
0
0
0
J + = −J2 + iJ1 , J − = J2 + iJ1 , J 3 = iJ3 , (i =
√
−1),
satisfy the commutation relations identical with (1.16). Thus we say that T3 is the complexification of E3 and E3 is a real form of T3 [6]. Due to this relationship between T3 and E3 , the abstract irreducible representation Q(ω, m0 ) of T3 [10] induces an irreducible representation of E3 . In this paper, we derive generating relations involving 3V2PTF Cn (x, y, z; τ1 , τ2 ) by using Lie-algebraic methods. In Section 2, we give a review of the basic properties of 3V2PTF Cn (x, y, z; τ1 , τ2 ) and their special cases. In Section 3, we derive generating relations involving 3V2PTF by using the representation Q(ω, m0 ) of the Lie algebra T3 . In Section 4, we obtain certain new and known generating relations involving various forms of Tricomi and Bessel functions. Finally, in Section 5, some concluding remarks are given. 2. Properties and special cases of 3V2PTF Cn (x, y, z; τ1 , τ2 ) The 3V2PTF Cn (x, y, z; τ1 , τ2 ) defined by Eqs. (1.8), (1.9) satisfy the following differential and pure recurrence relations: (2.1)
∂ Cn (x, y, z; τ1 , τ2 ) = −Cn+1 (x, y, z; τ1 , τ2 ), ∂x ∂ 1 Cn (x, y, z; τ1 , τ2 ) = − Cn+2 (x, y, z; τ1 , τ2 ), ∂y τ1 ∂ 1 Cn (x, y, z; τ1 , τ2 ) = − Cn+3 (x, y, z; τ1 , τ2 ), ∂z τ2
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∂ Cn (x, y, z; τ1 , τ2 ) = Cn−2 (x, y, z; τ1 , τ2 ) + ∂τ1 ∂ Cn (x, y, z; τ1 , τ2 ) = Cn−3 (x, y, z; τ1 , τ2 ) + ∂τ2
y Cn+2 (x, y, z; τ1 , τ2 ), τ12 z Cn+3 (x, y, z; τ1 , τ2 ) τ22
and (2.2) nCn (x, y, z; τ1 , τ2 ) = Cn−1 (x, y, z; τ1 , τ2 ) + xCn+1 (x, y, z; τ1 , τ2 ) + 2τ1 Cn−2 (x, y, z; τ1 , τ2 ) 3z Cn+3 (x, y, z; τ1 , τ2 ). τ2 The differential equation satisfied by 3V2PTF Cn (x, y, z; τ1 , τ2 ) is (2.3) ∂2 ∂ ∂2 ∂2 −x 2 − (1 + n) + 2τ1 + 3τ2 − 1 Cn (x, y, z; τ1 , τ2 ) = 0. ∂x ∂x ∂x∂τ1 ∂x∂τ2 + 3τ2 Cn−3 (x, y, z; τ1 , τ2 ) +
We note the following special cases of 3V2PTF Cn (x, y, z; τ1 , τ2 ): (1) (2.4)
Cn
x2 y 2 z 2 2yτ1 3zτ2 , , ; , 3 4 4 4 x2 x
=
x −n 2
Jn (x, y, z; τ1 , τ2 ),
where Jn (x, y, z; τ1 , τ2 ) is given by Eqs. (1.12), (1.13). (2) (2.5)
Cn
x2 y 2 z 2 2y 4z , , ; , 4 4 4 x2 x3
=
x −n 2
Jn (x, y, z),
where Jn (x, y, z) denotes 3-variable Bessel function (3VBF) defined by the generating function [4] (2.6) ∞ X x 1 y 2 1 z 3 1 n Jn (x, y, z)t = exp t− + t − 2 + t − 3 . 2 t 2 t 2 t n=−∞ (3) (2.7)
Cn (x, y, z; 1, 1) = Cn (x, y, z),
where Cn (x, y, z) denotes 3-variable Tricomi function (3VTF) defined by the generating function ∞ X x y z (2.8) Cn (x, y, z)tn = exp t − + t2 − 2 + t3 − 3 . t t t n=−∞ (4) (2.9)
Cn x, y, zτ2 2 ; τ1 → τ, τ2 → 0
= Cn (x, y; τ ),
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where Cn (x, y; τ ) denotes 2-variable 1-parameter Tricomi function (2V1PTF) defined by the generating function ([3]; p. 221, Eq.(9)). ∞ X x y (2.10) Cn (x, y; τ )tn = exp t − + τ t2 − 2 . t τt n=−∞ (5) (2.11)
2yτ x2 y 2 , , zτ2 2 ; 2 , τ2 → 0 4 4 x
Cn
=
x −n 2
Jn (x, y; τ ),
where Jn (x, y; τ ) denotes 2-variable 1-parameter Bessel functions (2V1PBF) defined by the generating function ([4]; p. 176 Eq.(1.2)) ∞ X x 1 y 1 (2.12) Jn (x, y; τ )tn = exp t− + τ t2 − . 2 t 2 τ t2 n=−∞ (6) (2.13)
Cn
x −n x2 y 2 2 2y , , zτ2 ; 2 , τ2 → 0 = Jn (x, y), 4 4 x 2
where Jn (x, y) denotes 2-variable Bessel functions (2VBF) defined by the generating function ([5]; p. 24 Eq.(1.8(a))) ∞ X x 1 y 2 1 n (2.14) Jn (x, y)t = exp t− + t − 2 . 2 t 2 t n=−∞ (7)
Cn x, y, zτ2 2 ; 1, τ2 → 0
(2.15)
= Cn (x, y),
where Cn (x, y) denotes 2-variable Tricomi functions (2VTF) defined by the generating function ∞ X x y (2.16) Cn (x, y)tn = exp t − + t2 − 2 . t t n=−∞ (8) (2.17)
Cn
x −n x2 2 2 , yτ1 , zτ2 ; τ1 → 0, τ2 → 0 = Jn (x), 4 2
where Jn (x) denotes ordinary Bessel function. (9) (2.18)
Cn x, yτ1 2 , zτ2 2 ; τ1 → 0, τ2 → 0
= Cn (x),
where Cn (x) denotes Tricomi function defined by the generating function (1.4).
TRICOMI FUNCTIONS
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3. Generating relations We derive the generating relations involving 3V2PTF Cn (x, y, z; τ1 τ2 ) by framing them into the context of the representation Q(ω, m0 ) of the Lie algebra T3 . We recall that Miller [10] have determined realizations of the irreducible representation Q(ω, m0 ) of T3 where ω, m0 ∈ C such that ω 6= 0 and 0 ≤ Re m0 < 1. The spectrum S of this representation is the set {m0 + k : k an integer} and the representation space V has a basis (fm )m∈S , such that (3.1)
J 3 fm = mfm , J + fm = ωfm+1 , J − fm = ωfm−1 , C0,0 fm = (J + J − )fm = ω 2 fm , ω 6= 0.
The commutation relations satisfied by the operators J 3 , J ± are (3.2)
[J 3 , J + ] = J + , [J 3 , J − ] = −J − , [J + , J − ] = 0.
In order to find the realizations of this representation on spaces of functions of two complex variables x and y, Miller ([10]; pp. 59–60) has taken the functions fm (x, y) = Zm (x)emy , such that relations (3.1) are satisfied for all m ∈ S, where the differential operators J 3 , J ± are given by ∂ J3 = , ∂y ∂ 1 ∂ J + = ey − , (3.3) ∂x x ∂y ∂ 1 ∂ J − = e−y − − . ∂x x ∂y In particular, we look for the functions (3.4)
fm (x, y, z, t; τ1 , τ2 ) = Zm (x, y, z; τ1 , τ2 )tm ,
such that (3.5)
K 3 fm = mfm , K + fm = ωfm+1 , K − fm = ωfm−1 , C0,0 fm = (K + K − )fm = ω 2 fm , (ω 6= 0; m ∈ S).
The set of operators {K 3 , K + , K − } satisfy the commutation relations identical to (3.2). There are numerous possible solutions of Eq. (3.5). We assume that the set of linear differential operators {K 3 , K + , K − } takes the form ∂ , ∂t ∂ (3.6) , K+ = t ∂x x ∂ 2τ1 ∂ 3τ2 ∂ ∂ K− = − + + − . t ∂x t ∂τ1 t ∂τ2 ∂t The operators in Eqs. (3.6) satisfy the commutation relations (3.2). K3 = t
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In terms of the functions Zm (x, y, z; τ1 , τ2 ) and using operators (3.6), relations (3.5) reduce to (i) ∂ Zm (x, y, z; τ1 , τ2 ) = ωZm+1 (x, y, z; τ1 , τ2 ), ∂x (ii) ∂ ∂ ∂ + 2τ1 −x + 3τ2 − m Zm (x, y, z; τ1 , τ2 ) = ωZm−1 (x, y, z; τ1 , τ2 ), ∂x ∂τ1 ∂τ2 (iii) (3.7)
∂2 ∂ ∂2 ∂2 −x 2 + 2τ1 + 3τ2 − (m + 1) Zm (x, y, z; τ1 , τ2 ) ∂x ∂x∂τ1 ∂x∂τ2 ∂x
= ω 2 Zm (x, y, z; τ1 , τ2 ). We can take ω = −1, without any loss of generality. For this choice of ω and in terms of the functions Zm (x), relations (3.1) become ([10]; p. 60 (3.25)) (i) m d − Zm (x) = −Zm+1 (x), dx x (ii) d m + Zm (x) = Zm−1 (x), dx x (iii) d2 1 d m2 (3.8) − 2− + 2 Zm (x) = Zm (x). dx x dx x We observe that (i) and (ii) of Eqs. (3.8) agree with the conventional recurrence relations for Bessel functions Jm (x) and (iii) coincides with the differential equation for Jm (x). Thus we see that Zm (x) = Jm (x) is a solution of Eqs. (3.8) for all m ∈ S. Similarly, we see that for ω = −1, (iii) of Eqs. (3.7) coincides with the differential equation (2.3) of 3V2PTF Cn (x, y, z; τ1 , τ2 ). In fact, for all m ∈ S the choice for Zm (x, y, z; τ1 , τ2 ) = Cm (x, y, z; τ1 , τ2 ) satisfy Eqs. (3.7). Thus we conclude that the functions fm (x, y, z, t; τ1 , τ2 ) = Cm (x, y, z; τ1 , τ2 )tm , m ∈ S form a basis for a realization of the representation Q(−1, m0 ) of T3 . By using ([10]; p. 18 (Theorem 1.10)), this representation of T3 can be extended to a local multiplier representation T ([10], p. 17) of T3 . Using operators (3.6), the local multiplier representation T (g), g ∈ T3 defined on F, the space of all functions analytic in a neighbourhood of the point (x0 , y 0 , z 0 , t0 , τ10 , τ20 ) = (1, 0, 0, 1, 1, 1), takes the form bt + (3.9) [T (exp bJ )f ](x, y, z, t; τ1 , τ2 ) = f x 1 + , y, z, t; τ1 , τ2 , x
TRICOMI FUNCTIONS
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c c [T (exp cJ − )f ](x, y, z, t; τ1 , τ2 ) = f x 1 − , y, z, t 1 − ; t t c −2 c −3 τ1 1 − , τ2 1 − , t t [T (exp aJ 3 )f ](x, y, z, t; τ1 , τ2 ) = f (x, y, z, tea ; τ1 , τ2 ). If g ∈ T3 is given by Eq. (1.14), we find T (g) = T (exp bJ + )T (exp cJ − )T (exp aJ 3 ) and therefore we obtain (3.10)
bt c c [T (g)f ](x, y, z, t; τ1 , τ2 ) = f x 1 + 1− , y, z, tea 1 − ; x t t bt c −2 c −3 τ1 1 − , τ2 1 − , < 1, t t x
c < 1. t
The matrix elements of T (g) with respect to the analytic basis (fm )m∈S are the functions Alk (g) uniquely determined by Q(−1, m0 ) of T3 and are defined by [T (g)fm0 +k ](x, y, z, t; τ1 , τ2 ) (3.11)
=
∞ X
Alk (g)fm0 +l (x, y, z, t; τ1 , τ2 ), k = 0, ±1, ±2, ±3, . . . .
l=−∞
Therefore, we prove the following result: Theorem 3.1. The following generating equation holds (3.12) c m c c −2 c −3 bt 1− Cm x 1 + 1− , y, z; τ1 1 − , τ2 1 − t x t t t ∞ |p| X (−1) = b(p+|p|)/2 c(−p+|p|)/2 0 F1 [−; |p| + 1; bc] Cm+p (x, y, z; τ1 , τ2 )tp , |p|! p=−∞ bt < 1, c < 1. x t Proof. Using (3.10), we obtain (3.13)
c m bt c c −2 c −3 Cm x 1 + 1− , y, z; τ1 1 − , τ2 1 − exp(mτ ) 1 − x t t t t ∞ X = Al,m−m0 (g)Cm0 +l (x, y, z; τ1 , τ2 )tm0 +l−m l=−∞
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and the matrix elements Alk (g) are given by ([10]; p. 56 (3.12)0 ), (3.14) Alk (g) = exp((m0 + k)a)
(−1)|k−l| (l−k+|k−l|)/2 (k−l+|k−l|)/2 b c 0 F1 [−; |k − l| + 1; bc], |k − l|!
valid for all integral values of l, k and where 0 F1 denotes confluent hypergeometric function [1]. Substituting the value of Alk (g) given by (3.14) into (3.13) and simplifying we obtain result (3.12). Corollary 3.1. The following generating equation holds (3.15) 1+
=
r −2 r −3 r m rνt r Cm x 1 + , y, z; τ1 1 + , τ2 1 + 1+ 2νt 2x 2νt 2νt 2νt
∞ X p=−∞
rνt < 1, (−ν)p Jp (r) Cm+p (x, y, z; τ1 , τ2 )tp , 2x
r < 1. 2νt
Proof. If bc 6= 0, we can introduce the co-ordinates r, ν such that b = rν 2 and r c = − 2ν , with these new co-ordinates the matrix elements (3.14) can be expressed as (3.16)
Alk (g) = exp((m0 + k)a) (−ν)l−k Jl−k (r), k = 0, ±1, ±2, . . .
and generating relation (3.12) yields (3.15).
4. Applications We discuss some applications of the generating relations obtained in the preceding section. I. Taking c = 0 and t = 1 in generating relation (3.12), we get ∞ X b (−b)p (4.1) Cm ((x + b), y, z; τ1 , τ2 ) = Cm+p (x, y, z; τ1 , τ2 ), < 1. p! x p=0 Again, taking b = 0 and t = 1 in generating relation (3.12), we get (1 − c)m Cm (x(1 − c), y, z; τ1 (1 − c)−2 , τ2 (1 − c)−3 ) (4.2)
=
∞ p X c p=0
p!
Cm−p (x, y, z; τ1 , τ2 ), |c| < 1. 2
2
2
II. Replacing x by x4 , y by y4 , z by z4 , τ1 by ( x2y2 )τ1 , τ2 by ( x3z3 )τ2 and t by xt 2 in generating relation (3.12) and using Eq. (2.4), we get (4.3) 1 − (2c/xt) m/2 1 + (2bt/x)
2c 1/2 2bt 1/2 1− Jm x 1 + , y, z; x xt
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2c −1 2bt 3/2 2c −3/2 2bt 1− , τ2 1 + 1− τ1 1 + x xt x xt ∞ |p| X (−1) = b(p+|p|)/2 c(−p+|p|)/2 0 F1 [−; |p| + 1; bc] Jm+p (x, y, z; τ1 , τ2 )tp , |p|! p=−∞ 2bt < 1, 2c < 1, x xt where Jm (x, y, z; τ1 , τ2 ) denotes the 3V2PBF defined by Eqs. (1.12) and (1.13). 2 2 2 Similarly replacing x by x4 , y by y4 , z by z4 , τ1 by ( x2y2 )τ1 , τ2 by ( x3z3 )τ2 and t xt by 2 in generating relation (3.15) and using Eq. (2.4), we get (4.4)
1 + (r/vxt) m/2
rvt 1/2 r 1/2 Jm x 1 + 1+ , y, z; 1 + (rvt/x) x vxt rvt r −1 rvt 3/2 r −3/2 τ1 1 + 1+ , τ2 1 + 1+ x vxt x vxt ∞ rvt r X = (−v)p Jp (r) Jm+p (x, y, z; τ1 , τ2 )tp , < 1, < 1. x vxt p=−∞
Further, taking t = τ1 = τ2 = 1 and b = −c in generating relation (4.3), we get 2c (4.5) Jm x 1 − , y, z x ∞ X (−1)3(|p|+p)/2 (|p|) 2 = c 0 F1 [−; |p| + 1; −c ] Jm+p (x, y, z), |p|! p=−∞
2c < 1, x
where Jm (x, y, z) denotes 3VBF given by Eq. (2.6). Similar result can be obtained from generating relation (4.4). III. Replacing z by zτ2 2 , τ1 by τ and then taking τ2 → 0 in generating relation (3.12) and using Eq. (2.9), we get c c −2 c m bt (4.6) 1− Cm x 1 + 1− , y; τ 1 − t x t t ∞ |p| X (−1) = b(p+|p|)/2 c(−p+|p|)/2 0 F1 [−; |p| + 1; bc] Cm+p (x, y; τ )tp , |p|! p=−∞ bt < 1, c < 1, x t
where Cm (x, y; τ ) denotes 2V1PTF given by Eq. (2.10).
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Further, replacing x by relation (4.6), we get (4.7)
=
1 − (2c/xt) 1 + (2bt/x)
m/2 Jm
x2 4 ,
y by
y2 4 ,
τ by
2yτ x2
and t by
xt 2
in generating
1/2 1/2 −1 ! 2bt 2c 2bt 2c x 1+ 1− , y; τ 1 + 1− x xt x xt
∞ X (−1)|p| (p+|p|)/2 (−p+|p|)/2 p b c 0 F1 [−; |p| + 1; bc] Jm+p (x, y; τ )t , |p|! p=−∞ 2bt < 1, 2c < 1, xt x
where Jm (x, y; τ ) denotes 2V1PBF given by Eq. (2.12). Further, taking t = τ = 1 and b = −c in generating relation (4.7), we get 2c (4.8) Jm x 1 − ,y x ∞ X 2c (−1)3(|p|+p)/2 (|p|) 2 < 1, = c 0 F1 [−; |p| + 1; −c ] Jm+p (x, y), |p|! x p=−∞ where Jm (x, y) denotes 2VBF given by Eq. (2.14). Similar results can be obtained from generating relation (3.15). IV. Replacing y by yτ1 2 , z by zτ2 2 and then taking τ1 , τ2 → 0 in generating relations (3.12) and (3.15) and using Eq. (2.18), we get c m bt c (4.9) 1− Cm x 1 + 1− t x t ∞ |p| X (−1) = b(p+|p|)/2 c(−p+|p|)/2 0 F1 [−; |p| + 1; bc] Cm+p (x)tp , |p|! p=−∞ bt < 1, c < 1 x t and (4.10)
r m rνt r 1+ Cm x 1 + 1+ 2νt 2x 2νt ∞ X rνt < 1, r < 1, (−ν)p Jp (r)Cm+p (x)tp , = 2x 2νt p=−∞
respectively, where Cm (x) denotes Tricomi function given by Eqs. (1.1) and (1.4).
TRICOMI FUNCTIONS
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Further, replacing x by z 2 /4, t by zt/2 in generating relation (4.9) and using Eq. (1.3), we obtain ([10]; p.62(3.29)), for Zm = Jm ) !m/2 1/2 1/2 ! 1 − 2c 2bt 2c zt (4.11) Jm z 1 + 1− z zt 1 + 2bt z =
∞ X (−1)|p| (p+|p|)/2 (−p+|p|)/2 p b c 0 F1 [−; |p| + 1; bc] Jm+p (z)t , |p|! p=−∞ 2bt < 1, 2c < 1. z zt
Several of the fundamental identities for cylindrical functions are special cases of generating relation (4.11). Also, for c = 0, t = 1 and b = 0, t = 1, relation (4.11) gives the formulas of Lommel ([10]; p. 62 (3.30) and (3.31) for Zm = Jm ). Again, replacing x by z 2 /4, t by z/2 in generating relation (4.10) and using Eq.(1.3), we obtain a generalization of Graf’s addition theorem ([10]; p. 63 (3.32), for Zm = Jm ) !m/2 r 1 + νz r 1/2 rν 1/2 (4.12) 1+ Jm z 1 + z νz 1 + rν z ∞ rν r X = (−ν)p Jp (r)Jm+p (z), < 1, < 1. z νz p=−∞ Further, taking c = 0, t = 1 and replacing x by x2 /4 and b by −xt/2 in relation (4.9) and using Eq. (1.3), we get a well known generating relation ([12]; p. 427, Eq. (56)). m/2 ∞ p p X t x (4.13) x2 − 2xt , m ∈ C Jm+p (x) = Jm p! x − 2t p=0 5. Concluding remarks We note that the expressions (3.11) are valid only for group elements g in a sufficiently small neighbourhood of the identity element of the Lie group T3 . However, we can also use operators (3.6) to derive generating relations for 3V2PTF and related functions with group elements bounded away from the identity. If f (x, y, z, t; τ1 , τ2 ) is a solution of the equation C0,0 f = ω 2 f , i.e., ∂2 ∂2 ∂ ∂2 (5.1) + 3τ2 − (m + 1) f (x, y, z, t; τ1 , τ2 ) −x 2 + 2τ1 ∂x ∂x∂τ1 ∂x∂τ2 ∂x = ω 2 f (x, y, z, t; τ1 , τ2 ),
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then the function T (g)f given by (3.10) satisfies the equation C0,0 (T (g)f ) = ω 2 (T (g)f ). This follows from the fact that C0,0 commutes with the operators K + , K − and K 3 . Now if f is a solution of the equation (5.2)
(x1 K + + x2 K − + x3 K 3 )f (x, y, z, t; τ1 , τ2 ) = λf (x, y, z, t; τ1 , τ2 )
for constants x1 , x2 , x3 and λ, then T (g)f is a solution of the equation (5.3)
[T (g)(x1 K + + x2 K − + x3 K 3 )T (g −1 )][T (g)f ] = λ[T (g)f ].
The inner automorphism µg of Lie group T3 defined by µg (h) = ghg −1 , h ∈ T3 ,
(5.4)
induces an automorphism µ?g of Lie algebra T3 where µ?g (α) = gαg −1 , α ∈ T3 . If α = x1 J + + x2 J − + x3 J 3 , where J + , J − and J 3 are given by Eq. (1.15) and g is given by Eq. (1.14), then we have (5.5)
µ?g (α) = (x1 ea − bx3 )J + + (x2 e−a + cx3 )J − + x3 J 3 ,
as a consequence of which, we can write (5.6) T (g)(x1 K + +x2 K − +x3 K 3 )T (g −1 ) = (x1 ea −bx3 )K + +(x2 e−a +cx3 )K − +x3 K 3 . To give an example of the application of these remarks, we consider the function f (x, y, z, t; τ1 , τ2 ) = Cm (x, y, z; τ1 , τ2 )tm , m ∈ C. Since C0,0 f = f and K 3 f = mf , so the function (5.7) [T (g)f ](x, y, z, t; τ1 , τ2 ) c c −2 c −3 ma m = e (t − c) Cm (x + bt) 1 − , y, z; τ1 1 − , τ2 1 − , t t t satisfies the equations (5.8)
C0,0 [T (g)f ] = [T (g)f ],
(5.9)
(−bK + + cK − + K 3 )[T (g)f ] = m[T (g)f ].
For a = b = 0 and c = −1, we can express the function (5.7) in the form (5.10) h(x, y, z, t; τ1 , τ2 ) −2 −3 ! x 1 1 m = (t + 1) Cm x+ , y, z; τ1 1 + , τ2 1 + , |t| < 1. t t t Now using the Laurent expansion ∞ X h(x, y, z, t; τ1 , τ2 ) = hk (x, y, z; τ1 , τ2 )tk , |t| < 1, k=−∞
TRICOMI FUNCTIONS
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in Eq. (5.8), we note that hk (x, y, z; τ1 , τ2 ) is a solution of differential equation (2.3) for each integer k. Since the function h(x, y, z, t; τ1 , τ2 ) is bounded for x = y = z = 0, we have hk (x, y, z; τ1 , τ2 ) = ck Ck (x, y, z; τ1 , τ2 ), ck ∈ C. Thus (5.11)
hk (x, y, z, t; τ1 , τ2 ) =
∞ X
ck Ck (x, y, z; τ1 , τ2 )tk .
k=−∞
Now from Eq. (5.9), we have (−K − + K 3 )h(x, y, z, t; τ1 , τ2 ) = mh(x, y, z, t; τ1 , τ2 ) and therefore it follows that ck+1 = (m − k)ck . Further taking x = y = z = 0 in (5.10) and using (5.11), we get c0 = 1/Γ(m + 1) and hence ck = 1/Γ(m − k + 1). Thus we obtain the following result: −2 −3 ! 1 1 x (5.12) (t + 1)m Cm , y, z; τ1 1 + , τ2 1 + x+ t t t =
∞ X Ck (x, y, z; τ1 , τ2 )tk , |t| < 1, Γ(m − k + 1)
k=−∞
which is obviously not a special case of generating relation (3.12). Several other examples of generating relations can be derived by this method, see for example Weisner [17]. We have considered 3V2PTF Cm (x, y, z; τ1 , τ2 ) within the group representation formalism. These functions appeared as basis functions for a realization of the representation Q(−1, m0 ) of the Lie algebra T3 . The analysis presented in this paper confirms the possibility of extending this approach to other useful forms of generalized Tricomi functions as well as to their Bessel counter parts. References [1] L. C. Andrews, Special Functions for Engineers and Applied Mathematician, Macmillan Company, New York, 1985. [2] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, Advanced special functions and applications, (Melfi, 1999), 147–164, Proc. Melfi Sch. Adv. Top. Math. Phys. 1, Aracne, Rome 2000. [3] G. Dattoli, M. Migliorati, and H. M. Srivastava, Some families of generating functions for the Bessel and related functions, Georgian Math. J. 11 (2004), no. 2, 219–228. [4] G. Dattoli and A. Torre, Theory and Applications of Generalized Bessel Functions, ARACNE, Rome, Italy, 1996. [5] G. Dattoli, A. Torre, S. Lorenzutta, G. Maino, and C. Chiccoli, Theory of generalized Bessel functions. II, Nuovo Cimento B (11) 106 (1991), no. 1, 21–51.
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[6] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. [7] S. Khan, Harmonic oscillator group and Laguerre 2D polynomials, Rep. Math. Phys. 52 (2003), 227–233. [8] S. Khan and M. A. Pathan, Lie theoretic generating relations of Hermite 2D polynomials, J. Comput. Appl. Math. 160(2003), no. 1-2, 139–146. [9] S. Khan, G. Yasmeen, and A. Mittal, Representation of Lie algebra T3 and 2-variable 2-parameter Bessel functions, J. Math. Anal. Appl. 326 (2007), no. 1, 500–510. [10] W. Jr. Miller, Lie Theory and Special Functions, Academic Press, New York and London, 1968. [11] E. D. Rainville, Special Functions, Macmillan, Reprinted by Chelsea Publ. Co., Bronx, New York, 1971. [12] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Limited, Chichester, 1984. [13] N. Ja. Vilenkin, Special Functions and the Theory of Group Representations, Amer. Math. Soc. Trans. Providence, Rhode Island, 1968. [14] A. Wawrzy¯ nczyk, Group Representation and Special Functions, PWN-Polish Scientific Publ., Warszawa, 1984. [15] L. Weisner, Group-theoretic origin of certain generating functions, Pacific J. Math. 5 (1955), 1033–1039. [16] , Generating functions for Hermite functions, Canad. J. Math. 11 (1959), 141– 147. [17] , Generating functions for Bessel functions, Canad. J. Math. 11 (1959), 148–155. Subuhi Khan Department of Mathematics Aligarh Muslim University Aligarh-202002, India E-mail address:
[email protected] Mumtaz Ahmad Khan Department of Applied Mathematics Faculty of Engineering Aligarh Muslim University Aligarh–202002, India Rehana Khan Department of Applied Mathematics Faculty of Engineering Aligarh Muslim University Aligarh–202002, India
