Georgian Mathematical JournalVolume 14 (2007), Number 1, 99–107
GENERATING RELATIONS OF HERMITE–TRICOMI
GIUSEPPE DATTOLI, SUBUHI KHAN, AND GHAZALA YASMIN
Abstract. Motivated by recent studies of the properties of new classes ofpolynomials constructed in terms of quasi-monomials, certain generating re-lations involving Hermite–Tricomi functions are obtained. To accomplishthis we use the representation Q(w, m0) of the 3-dimensional Lie algebra T3. Some special cases are also discussed.
2000 Mathematics Subject Classification: 33C45, 33C50, 33C80. Key words and phrases: Generating relations, Lie algebra, Hermite–Tricomi functions.
The notion of quasi-monomiality [2, 8] has been exploited within different
contexts to deal with isospectral problems [2, 8] and to study the properties ofnew families of special functions [2]. The concept of quasi-monomiality can besummarized as follows:
(b) Let fn(x), n ∈ N, x ∈ C, be a polynomial. fn(x) is said to be a quasi-
Mfn(x) = fn+1(x), ˆ
P fn(x) = nfn−1(x).P are recognized as multiplicative and derivative op-
erators respectively. Furthermore, if f0(x) = 1, from the first of equation (1.1)it follows that
Mn(1) = fn(x).
We note that the two variable Hermite Kamp´e de F´eri´et polynomials
ISSN 1072-947X / $8.00 / c Heldermann Verlag www.heldermann.de
are quasi-monomials under the action of the operators
Recently, Dattoli and Torre [3] studied the properties of new classes of poly-
nomials constructed in terms of quasi-monomials. We consider the Hermite–Tricomi functions (HTF) HCn(x, y) defined by the series ([3]; p. 24 (36 b))
and the generating function for HCn(x, y) is given as ([3]; p. 25 (47))
tnHCn(x, y) = exp t −
We note that for y = 0, the HTF HCn(x, y) reduces to the Tricomi functions
Cn(x), which are linked to the ordinary Bessel functions by the relation ([3]; p. 25 (469))
Cn(x) = x−n2 Jn(2 x).
The HTF HCn(x, y) satisfy the following differential and pure recursion rela-
∂x HCn(x, y) = −HCn+1(x, y),∂y HCn(x, y) = HCn+2(x, y),nHCn(x, y) = HCn−1(x, y) + x HCn+1(x, y) − 2y HCn+2(x, y).
In this paper we obtain generating relations involving HTF HCn(x, y). To
accomplish this we use the representation Q(w, m0) of the 3-dimensional Liealgebra T3. We also consider some special cases which would inevitably yieldnew generating relations involving Tricomi functions and a few known resultsof Miller [6] including Graf’s addition theorem.
2. Representation Q(w, m0) of T3 and HTF HCn(x, y)
We have the following isomorphism ([6]; p. 36)
= L(T3) ⊕ (E),
where (E) is the 1-dimensional Lie algebra generated by E. The nontrivial partof the representation theory of G(0, 0) is concerned with the subalgebra T3, theLie algebra of the 3-dimensional complex local Lie group T3 ([6]; p. 10). The
GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
matrix group T3 is the set of all 4 × 4 matrices
The group T3 has the topology of C3 and is simply connected ([7]; Ch. 8). A basis for T3 = L(T3) is provided by the matrices
[J 3, J ±] = ±J ±,
Also, we note that the Lie algebra E3 of the Euclidean group in the plane
E3 is the real form of T3 or T3 is the complexification of E3([5]; p. 152). Dueto this relationship between T3 and E3, the abstract irreducible representationQ(w, m0) of T3 induces an irreducible representation of E3.
Consider the irreducible representation Q(w, m0) of T3, where w, m0 ∈ C
such that w = 0 and 0 ≤ Re m0 < 1. The spectrum S of this representationis the set {m0 + n : n an integer }. In particular, we look for the functionsfn(x, y, t) = Zn(x, y)tn such that
J+fn = wfn+1,C0,0fn = (J+J−)fn = w2fn,
for all n ∈ S, where the differential operators J±, J3 take the form
and note that the commutation relations of these operators are identical with(2.3).
Without any loss of generality, we can assume that w = 1. In terms of the
functions Zn(x, y) = HCn(x, y), relations (2.4) become
∂xHCn(x, y) = −HCn+1(x, y),H Cn(x, y) = H Cn−1(x, y),∂x HCn(x, y) = −HCn(x, y),
where HCn(x, y) is given by (1.6)–(1.7).
The functions fn(x, y, t) = HCn(x, y)tn, n ∈ S, form a basis for the realization
of the representation Q(1, m0) of T3. We will extend this representation of T3,to a multiplier representation of T3. According to Miller ([6]; p. 18 (Theorem1.10)), the differential operators given by (2.5) generate a Lie algebra which isthe algebra of generalized Lie derivatives of a multiplier representation T (g) ofT3 acting on the space F of all functions analytic in some neighbourhood of thepoint (x0, y0, t0) = (1, 1, 1).
A simple computation using equations (2.5) gives
[T (exp bJ +)f ](x, y, t) = f x 1 −
[T (exp cJ −)f ](x, y, t) = f x 1 +
[T (exp τ J 3)f ](x, y, t) = f (x, y, t exp τ ),
for f ∈ F and |b|, |c|, |τ | sufficiently small. If g ∈ T3 is given by equation (2.1),we find
g = (exp bJ +)(exp cJ −)(exp τ J 3),
and therefore the multiplier representation takes the form
[T (g)f ](x, y, t) = f x 1 −
The matrix elements Alk(g) of T (g), with respect to the basis fn are uniquely
determined by Q(1, m0) and are defined by
lk(g)fm0+l(x, y, t),k = 0, ±1, ±2, . . . .
Therefore, we can prove our main result.
GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
Theorem 1. The following generating equation holdsc(−n+|n|)/2b(n+|n|)/2
0F1(−; |n| + 1; bc)H Cm+n(x, y)tn,H Cm0+l(x, y)tm0+l,
and the matrix elements Alk(g) are given by ([6]; p. 56 (3.12))
0 + k)τ ) c(k−l+|k−l|)/2 b(l−k+|k−l|)/2
× 0F1(−; |k − l| + 1; bc),
valid for all integral values of l and k.
Substituting the value of Alk(g) given by (2.12) into (2.11) and simplifying,
If bc = 0, we can introduce the coordinates r, v in place of b, c by r = (ibc)1/2
and v = (b/ic)1/2 such that b = rv/2, c = −r/2v. In terms of the coordinatesr, v the matrix elements are given by ([6]; p. 56 (3.14))
Alk(g) = exp ((m0 + k)τ)(−v)l−kJ l−k(−r),
(−v)nJn(−r)HCm+n(x, y)tn,
which can be viewed as a generalization of Graf’s addition theorem ([4]; p. 44).
We consider some special cases of generating relations given by (2.10) and
I. Taking c = 0 and t = 1 in relation (2.10), we obtain
n! HCm+n(x, y),
Again taking b = 0 and t = 1 in relation (2.10), we obtain
(1 + c)mHCm x(1 + c), y(1 + c)2 =
n! HCm−n(x, y).
II. Taking y = 0 in relations (2.10) and (2.13), we obtain the following generat-ing relations of the Tricomi functions
c(−n+|n|)/2b(n+|n|)/2
× 0F1(−; |n| + 1; bc)Cm+n(x)tn,
(−v)nJn(−r)Cm+n(x)tn,
III. Replacing x by z2/4, t by −zt/2 and using relation (1.8) in relation (3.3),we obtain ([6]; p. 62 (3.29), for Zm = Jm)
c(−n+|n|)/2b(n+|n|)/2
0F1(−; |n| + 1; bc)Jm+n(z)tn,
Several of the fundamental identities for cylindrical functions are special cases
of relation (3.5). Also, for c = 0, t = 1 and b = 0, t = 1 relation (3.5) gives theformulas of Lommel ([6]; p. 62 (3.30) and (3.31)).
Again replacing x by z2/4, t by −z/2 and using relation (1.8) in relation (3.4)
we obtain ([6]; p. 63 (3.32)), which is a generalization of Graf’s addition theorem([4]; Vol. II, p. 44).
GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
In this concluding section, we will show how the already obtained results can
be further generalized by means of operational formalism.
To this end, we remind that the HTF can be obtained from the ordinary
Tricomi functions just by using the identity
H Cn(x, y) = ey ∂2
The above relation implies that any identity valid for Cn(x) can be extended
to HTF by means of simple algebraic manipulations. Furthermore, (4.1) alsoensures that HCn(x, y) satisfies the partial differential equation
∂y HCn(x, y)) = ∂x2 HCn(x, y)),H Cn(x, 0) = Cn(x).
Just to give an idea of how the method works we note that the operators
J±,3corresponding to the ordinary case are specified by
while those relevant to the functions HCn(x, y) will be provided by
It is evident that the only operator affected by such transformation is J−
which does not commute with ∂ and thus we find
which, taking into account equation (4.2), coincides with the definition given in(2.5).
According to the above discussion we can try a further generalization of the
previous results. Indeed, we can define the function
∞ (−1)rHr(x1, x2, x3)
H Cn(x1, x2, x3) =
n−3r(x1, x2)
n(x1, x2, x3) = n!
As it is easy to see, function (4.6) is specified by the operational identity
H Cn(x1, x2, x3) = e
1 (H Cn(x1, x2)).
It likewise easily can be verified that the function HCn(x1, x2, x3) satisfies the
∂x HCn(x1, x2, x3)) =
H Cn(x1, x2, x3)),∂x HCn(x1, x2, x3)) =
H Cn(x1, x2, x3)).
The relevant operator J− can therefore be written as
and, furthermore, we can easily generalize the previous identities. From (2.10)
we find, after multiplying both sides by the operators e 3 ∂x31 (x1 ≡ x, x2 ≡ y),that
e(−n+|n|)/2b(n+|n|)
0F1(−; |n| + 1; bc)H Cm+n(x1, x2, x3)tn.
Along the same lines we can obtain the generalization to the m variable case
n−mr (x1, . . . , xm−1) ,
1, x2, . . . , xm) = n!
as will be discussed in a forthcoming paper.
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GENERATING RELATIONS OF HERMITE–TRICOMI FUNCTIONS
8. Yu. Smirnov and A. Turbiner, Hidden sl2-algebra of finite-difference equations. Mod.
(Received 11.04.2006; revised 29.06.2006)
G. Dattoli and S. KhanENEA, Gruppo Fisica Teorica e Matematica ApplicataUnita Tecnico Scientifica Tecnologie Fisiche AvanzateCentro Ricerche FrascatiC.P. 65, Via Enrico Fermi 45, I- 00044 Frascati, RomeItalyE-mail: [email protected]
S. Khan (Permanent address) and G. YasminDepartment of MathematicsAligarh Muslim UniversityAligarh-202 002, India

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