factorization of differential operators for some special functions

SOOCHOW JOURNAL OF MATHEMATICS

Volume 29, No. 2, pp. 147-156, April 2003

FACTORIZATION OF DIFFERENTIAL OPERATORS FOR SOME SPECIAL FUNCTIONS BY ANGELA BERNARDINI AND PIERPAOLO NATALINI Abstract. Let {Sn (x)}∞ n=0 be a sequence of special functions depending on a parameter n, and suppose that there exist two sequences of differential operators En− and En+ satisfying the following properties En− (Sn (x)) = Sn−1 (x), En+ (Sn (x)) = Sn+1 (x). By constructing these two operators, and using the factorization method ([10]), we determine the differential equations satisfied by some special functions, including some new classes recently introduced by G. Dattoli et al. by using the so called Monomiality principle [4]-[6].

1. Introductory Definitions Let {Sn (x)}∞ n=0 be a sequence of special functions depending on a parameter

n. For n = 0, 1, 2, . . . , we consider two sequences of operators En− and En+ satisfying the following properties En− (Sn (x)) = Sn−1 (x), En+ (Sn (x)) = Sn+1 (x).

En− and En+ play the role analogous to that of derivative and multiplicative operators respectively on monomials. In recent papers the monomiality principle and Received August 19, 2001; revised September 28, 2002. AMS Subject Classification. 33C45, 33C55. Key words. differential operators, factorization method, special functions, differential equations. 147

148

ANGELA BERNARDINI AND PIERPAOLO NATALINI

the associated operational rules were used to explore new classes of isospectral problems leading to non trivial generalizations of special functions (see e.g. [1][5]-[2]-[3]). Most of the properties of functions associated with En− and En+ can be deduced by using operational rules with these two operators. The operators defined in this paper are varying with the above mentioned parameter n. The iterations of En− and En+ to Sn (x) give the following relations: − (En+1 En+ )Sn (x) = Sn (x), + En− )Sn (x) = Sn (x), (En−1 − En− )Sn (x) = S0 (x), (E1− E2− · · · En−1 + + En−2 · · · E1+ E0+ )S0 (x) = Sn (x). (En−1

These operational relations will be used to derive differential equations, in general of higher order, satisfied by some special functions. The classical factorization method introduced in [10] was applied to construct second-order differential equations. − and E + 2. Operators En n

In the following we will use special functions satisfying a (d + 2)-term recurrence relation with d ≥ 1. The number of the terms of recurrence is fixed and greater than or equal to 3. There are many classes of special functions with this type of recursion formulas such as orthogonal polynomials and confluent hypergeometric functions (see e.g. [9]). The following main result, proved in [9], will be used: Theorem 2.1. Let {Sn (x)}∞ n≥0 be a sequence of functions satisfying the following (d + 2)-term recurrence relation Sn+d+1 (x) = (x − αn+d )Sn+d (x) +

d

βn+d−k Sn+d−k (x),

(2.1)

k=1

with some proper initial conditions. If there is a differential operator En− such that En− Sn (x) = Sn−1 (x),

(2.2)

FACTORIZATION OF DIFFERENTIAL OPERATORS

149

then the functions of the sequence {Sn (x)}∞ n≥0 satisfy the following differential equation − En+ )Sn (x) = Sn (x), (n ≥ d), (2.3) (En+1 where + = (x − αn+d ) + En+d

d

βn+d−k

k

− En+d−k+j .

(2.4)

j=1

k=1

If there is an operator En+ such that En+ Sn (x) = Sn+1 (x),

(2.5)

− , ∀n ≥ 0, is given by then the operator En+1



− En+1





d d−1 d−1 d−k−1  1  + + +  = En+k − (x − αn+d ) En+k + βn+d−k En+j .  βn k=1 j=1 k=1 k=1

(2.6) The authors, in their paper [9], apply the above stated result to the classical orthogonal polynomials, d-orthogonal polynomials, confluent hypergeometric functions and hypergeometric functions. In the following section we illustrate the factorization method for some other classes of special functions. In each case, the operators En− and En+ will be given explicitly. 3. Applications 3.1. Bessel functions of first kind Let Jp (z) be the Bessel function of first kind of order p Jp (z) =

p ∞ z

2

(−1)k k!Γ(p + k + 1) k=0

2k

z 2

.

Then we have the following result Theorem 3.1. The functions z p Jp (z) and z −p Jp (z) satisfy the following differential equations

d2 d z 2 + (1 − 2p) + z (z p Jp (z)) = 0, dz dz

(3.1)

150


d2 d z 2 + (1 + 2p) − z (z −p Jp (z)) = 0 dz dz

(3.2)

respectively. Proof. First we consider the function z p Jp (z). Since it is easy to verify the relation 1 d p [z Jp (z)] = z p−1 Jp−1 (z), z dz we can define the operator Ep− in the following way Ep− =

1 d . z dz

Furthermore they satisfy the recurrence relation 2p(z p Jp (z)) = z 2 (z p−1 Jp−1 (z)) + z p+1 Jp+1 (z). By using the Theorem 2.1 we can write Ep+ = 2p − z

d . dz

Then the eq. (3.1) immediately follows from the factorization (2.3). In an analogous way, for z −p Jp (z), we find the operator Ep+ = −

1 d . z dz

Furthermore they satisfy the recurrence relation 2p(z −p Jp (z)) = z −(p−1) Jp−1 (z) + z 2 (z −(p+1) Jp+1 (z)). By using the Theorem 2.1 we can write Ep− = 2p + z

d . dz

Then the eq. (3.2) immediately follows from the factorization (2.3). Remark 3.1. Since we will use always the same method, which is based on Theorem 2.1, in the following we will limit ourselves to derive explicitly the operators E∗− and E∗+ by means of which the searched differential operator can be factorized.


151

3.2. Bessel functions of second kind Let Yp (z) be the Bessel function of second kind of order p Yp (z) =

Jp (z) cos(pπ) − (−1)p Jp (z) . sin(pπ)

Then we have the following result Theorem 3.2. The function z p Yp (z) satisfies the following differential equa

tion

d2 d z 2 + (1 − 2p) + z (z p Yp (z)) = 0. dz dz

(3.3)

Proof. By a similar procedure, we can derive Eq. (3.3) from the factorization (2.3) where 1 d . Ep− = z dz and d Ep+ = 2p − z . dz 3.3. Bessel functions of third kind (1)

Let Hp (z) be the Bessel function of third kind (or Henkel function) of order p Hp(1) (z) = Jp (z) + iYp (z). Then we have the following result. (1)

Theorem 3.3. The function z p Hp (z) satisfies the following differential equation d2 d z 2 + (1 + 2p) + z (z p Hp(1) (z)) = 0. (3.4) dz dz Proof. We find, in this case: Ep− =

1 d , z dz

and Ep+ = 2p − z

d . dz

152


Then the eq. (3.4) immediately follows from the factorization (2.3). 3.4. Modified Laguerre polynomials These polynomials are defined by the generating function (see [13], Eq. 8.4 (46), p.425) −α xt

(1 − t)

e

=

∞

fnα(x)tn = Gα (x, t),

n=0 (α)

and are connected with the classical Laguerre polynomials Ln (x) or the PoissonCharlier polynomials cn (x; α) by means of the formula (see [13], Eq. 8.4 (45), p.425) xn cn (α; x). n! The explicit expression for these functions is given by (x) = fn−α(x) = (−1)n L(α−n) n

fnα (x) =

n xk Γ(n − k + α) k=0

k! (n − k)! Γ(α)

(3.5)

,

while their integral representation is given by fnα (x)

1 = n! Γ(α)

∞ 0

e−ξ ξ α−1 (x + ξ)n dξ.

We can prove the following result. Theorem 3.4. The modified Laguerre polynomials satisfy the following differential equation

d2 d + 1 fnα (x) = 0. x 2 − (x + α) dx dx

(3.6)

Proof. Since these polynomials belong to the Appell class, we have En− =

d . dx

Furthermore they satisfy the recurrence relation (n = 0), f1α (x) = (x + α)f0α (x), 1 α α [n + (x + α)]fnα (x) − xfn−1 (x) = (x), fn+1 n+1

(n = 1, 2, . . .).


153

By using the Theorem 2.1 we can write En+

1 d = n + (x + α) − x . n+1 dx

Remark 3.2. Note that using formula (3.5) we can rewrite Eq. (3.6) in terms of the more familiar L-notation of the classical Laguerre polynomials, that is,

d2 d x 2 − (x − α) (x) = 0. + 1 L(α−n) n dx dx

3.5. Generalized Hermite-type truncated exponential polynomials Starting by the (Gould-Hopper) generalized Hermite polynomials (see [13], Eq. 1.9 (6), p.76) [n]

gnm (x, h) =

m

n! hk xn−mk , k! (n − mk)! k=0

where m is a positive integer, we can define the following polynomials [n]

[m] hn (x)

m

xn−mk , (n − mk)! k=0

=

called generalized Hermite-type truncated exponential polynomials. For these polynomials we have the integral representation 1 [m] hn (x) = n!

∞ 0

e−ξ gnm (x, ξ)dξ,

and, by means of the formula ∞ n t n=0

n!

we obtain the generating function of ∞

m

gnm (x, h) = ext+ht , [m] hn (x)

tn ([m] hn (x)) =

n=0

We can prove the following result.

ext = Gm (x, t). 1 − tm

154


Theorem 3.5. The Generalized Hermite-type truncated exponential polynomials satisfy the following differential equation

dm+1 dm d x m+1 − n m − x +n dx dx dx

[m] hn (x)

= 0.

(3.7)

Proof. In this case the operator En− is given by En− =

d , dx

while the recurrence relation is given by [m] hn+1 (x)

=

1 (n + 1)[m] hn−m+1 (x) + x[m] hn (x) − x[m] hn−m (x) . n+1


dm−1 x dm = m−1 + 1− m . dx n+1 dx

3.6. An extension of the Laguerre-type polynomials Let ln (x) be the so called Laguerre-type troncated exponential polynomials, defined by ln (x) =

n (−1)k xk k=0

(k!)2

.

A generalization of these polynomials is given by the following new class of polynomials ln,α (x) =

n

(−1)k xk , k!Γ(k + α + 1) k=0

where α is a real number. By means of the associate Laguerre polynomials L(α) n (x, y) =

n (−x)k Γ(n + α + 1)y n−k k=0

(n − k)!k!Γ(α + k + 1)

,

and their generating function ∞

tn yt L(α) n (x, y) = e Cα (xt), Γ(n + α + 1) n=0


155

where Cα (xt) is the Tricomi-Bessel function Cα (xt) =

∞ (−1)k xk tk k=0

k! (α + k)!

,

we can write the integral representation 1 ln,α (x) = Γ(n + α + 1)

∞ 0

e−ξ L(α) n (x, ξ)dξ,

and the generating function ∞

tn ln,α (x) =

n=0

1 Cα (xt). 1−t

Theorem 3.6. The polynomials ln,α (x) satisfy the following differential equation

3 d2 d4 2 d + x (α + 6) + x [x + α + 1 + (α + 1)(α + 2) + 2(α + 2) dx4 dx3 dx2 d [(α + 1)2 + x(α + 1) − (n + 1)(α + 1)(n + α + 1)+x] −(n + 1)(n + α + 1)] + dx

x3

+(α + 1) + (n + 1)(n + α + 1) ln,α (x) = 0.

(3.8)

Proof. For these polynomials we have En− = −

1 d α+1 d x . xα dx dx

Furthermore we have the following recurrence relation ln+1,α (x) =

1 ([(n + 1)(n + α + 1) − x]ln,α (x) + xln−1,α (x)) . (n + 1)(n + α + 1)


x 1 d d =1− 1 + α xα+1 . (n + 1)(n + α + 1) x dx dx

Acknowledgment We should like to thank the anonymous referee for his useful remarks and suggestions.

156


References [1] G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, in the Proceedings of the International Workshop on “Advanced Special Functions and Their Applications”, Melfi (PZ) Italy, May 12-15, 1999, ed. by D. Cocolicchio, G. Dattoli and H. M. Srivastava, Aracne Editrice, Rome, 1999, 147-164. [2] G. Dattoli, Generalized polynomials, operational identities and applications in pure and applied mathematics, J. Comput. Appl. Math., 118(2000), 111-123. [3] G. Dattoli, C. Cesarano and D. Sacchetti, A note on the monomiality principle and generalized polynomials, Rend. Mat., (to appear). [4] G. Dattoli, C. Cesarano and D. Sacchetti, A note on truncated polynomials, Appl. Math. Comp., (to appear). [5] G. Dattoli, S. Lorenzutta, A. M. Mancho and A. Torre, Generalized polynomials and associated operational identities, J. Comput. Appl. Math., 108(1999), 209-218. [6] G. Dattoli, P. E. Ricci and L. Marinelli, Generalized truncated exponential polynomials and applications, Rend. Ist. Mat. Univ. Trieste, (to appear). [7] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980. [8] M. X. He and P. E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math., 139(2002), 231-237. [9] M. X. He and P. E. Ricci, Differential equations of some classes of special functions via the factorization method, J. Comput. Anal. and Appl., (to appear). [10] L. Infeld and T. E. Hull, The factorization method, Rev. Mod. Phys., 23(1951), 21-68. [11] P. Maroni, L’orthogonalit´ e et les récurrences de polynˆ omes d’ordre sup´ erieur a ` deux, Ann. Fac. Sci. Toulouse, 10:1(1989), 105-139. [12] A. Nikiforov and V. Uvarov, Special Functions of Mathematical Physics, Birkh¨ auser, Basel, Boston, 1988. [13] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Wiley, New York, 1984. Departamento de Fisica y Matem´ atica Aplicada, Universidad de Navarra, c/ Irunlarrea s/n, Edificio Los Casta˜ nos, E-30080 Pamplona, Espa˜ na. E-mail: angela@obelix.fisica.unav.es Dipartimento di Matematica, Universit` a degli Studi Roma Tre, Largo San Leonardo Murialdo, 1 00146 - Roma, Italia. E-mail: [email protected]