INFINITE INTEGRALS AND OPERATIONAL METHODS
arXiv:1105.5967v1 [math.CA] 30 May 2011
´ D. BABUSCI, G. DATTOLI, G. H. E. DUCHAMP, K. GORSKA, AND K. A. PENSON Abstract. An operatorial method, already employed to formulate a generalization of the Ramanujan master theorem, is applied to the evaluation of integrals of various type. This technique provide a very flexible and powerful tool yielding new results encompassing various aspects of the special function theory.
The use of operational methods is, sometimes, very much useful to “guess” specific theorems in various fields of analysis. The drawback of such an approach is that any of its consequences should be validated by a more rigorous procedure. This is indeed the case of the Ramanujan master theorem [1], according to which if a function f (x) can be expanded around x = 0 in a power series of the form ∞ X ϕ(n) f (x) = (−x)n , (1) n! n=0
with ϕ(0) = f (0) 6= 0, then
Z
∞
dx xν−1 f (x) = Γ(ν) ϕ(ν) .
(2)
0
The proof of this identity has been obtained in Refs. [2, 3, 4] by setting f (x) =
∞ X cˆ n
n=0
n!
(−x)n ϕ(0) = e−ˆc x ϕ(0) ,
(3)
with cˆ being an umbral operator such that [5] cˆ n ϕ(0) = ϕ(n) .
(4)
The theorem has been exploited in Refs. [3, 4] in a generalized version to obtain, in a very simple way, old and new formulae for integrals and successive derivatives of Bessel functions. The noticeable feature emerging from such a procedure is that cylindrical Bessel functions can be formally treated as ordinary gaussians. For example, the n-th order cylindrical Bessel function can be written as 1 2 ϕ(n) = Jn (2x) = (ˆ c x) n e−ˆc x ϕ(0) . (5) n! According to the above restyling, and by taking the freedom of treating cˆ as an ordinary constant, an interesting plethora of new results concerning integrals of Bessel functions or their derivatives can be obtained. Just to give an example of how the method works, we consider the integral Z ∞ 2 (α > 0, β > 0) . (6) dx x J2 ν (α x) e i β x Bν (α, β) = 0
2000 Mathematics Subject Classification. Primary 33. Key words and phrases. Integrals, Ramanujan’s master theorem, Bessel functions, Struve function, Hermite polynomials. 1
The use of eq.(5) allows to treat this integral as an elementary exponential integral and indeed, under the hypothesis α2 < 4 β, we find 1 α 2 ν i ν+1 α2 bν −i Bν (α, β) = (7) 2 2 β 4β where we have introduced the function
bν (x) =
∞ X Γ(ν + k + 1) xk . Γ(2 ν + k + 1) k! k
In the case ν = 0, eq.(7) gives α2 i exp −i , B0 (α, β) = 2β 4β
(8)
in agreement with eqs. 6.728.3 and 6.728.4 of Ref. [6], and eq. 2.12.18.7 of Ref. [7]. We consider now the Struve function [8] Hν (x) =
x 2 k+ν+1 (−1)k , Γ(k + 32 ) Γ(k + ν + 32 ) 2 k=0
∞ X
(9)
that can be rewritten as Hν (x) = with
x ν+1 2
2 ˆν x exp −h ϕ(0) 2
ˆ ν )n ϕ(0) = ϕ(n) = (h
Γ(n + 1) . Γ(n + + ν + 32 ) 3 2 ) Γ(n
By applying our method, and using the Euler’s reflection formula, we find Z ∞ 1 Γ(1 + ν2 ) Γ(− ν2 ) dx Hν (b x) = 1+ν b Γ( 1−ν 0 2 ) Γ( 2 ) 1 . (−2 < Re ν < 0, b > 0) , =− b tan π2ν that is formula 6.811.1 of Ref. [6], and, for example √ Z ∞ Hν (x) π π ˆ −1/2 dx ν+1 = ν (h ϕ(0) = ν ν) x 2 2 Γ(ν + 1) −∞
(10)
(11)
(12)
(13)
(see formula 6.813.2 of Ref. [7]). It is evident that the method can easily be applied also to more complicated integrals involving the Struve functions. As a further proof of the uselfuness of the method, we apply it to the derivation of the following generating function ∞ n X t Jmn (2x) , (14) G(x, t|m) = n! n=0
that, taking into account eq.(5), can be written as G(x, t|m) = exp (ˆ c x)m t − cˆ x2 ϕ(0) . 2
(15)
If we introduce the higher-order Hermite polynomials [9] [n/m]
un−m k v k , (n − m k)! k!
(16)
Hn(m) (u, v) = exp (u z + v z m )
(17)
Hn(m) (u, v)
X
= n!
k=0
their generating function ∞ X zn
n=0
n!
allows us to recast eq.(15) in the form G(x, t|m) =
∞ X cˆ n n=0
n!
Hn(m) (−x2 , xm t) ϕ(0) .
(18)
Furthermore, as a consequence of the definitions (5) and (16), it is possible to show that (m)
(x2 , (−x)m t)
(19)
(−1)k (m) (x, y) , H k! (n + k)! k
(20)
G(x, t|m) = H C0 where (m) H Cn (x, y)
=
∞ X k=0
are the so-called Hermite-based Tricomi functions [9]. The correctness of the above identity has been checked numerically. We remark that its derivation with conventional means is extremely more involved. In the same spirit, we will use operational methods to evaluate various families of integrals. In particular, we consider the integrals of the type Z ∞ dt f (x g(t)) . (21) I(x) = −∞
The identity [10]
[g(t)] x ∂x f (x) = f (x g(t)) can be used to write I(x) =
Z
(22)
∞
dt [g(t)] x ∂x f (x) ,
(23)
−∞
and, by assuming that the integral (a = constant) Z ∞ dt [g(t)] a F (a) = −∞
exists, one can express I(x) as
I(x) = Fˆ (x ∂x ) f (x) , If the function f (x) admits a power series expansion f (x) =
∞ X
α(k) x m k+p ,
(24)
(25)
k=0
as a consequence of the identity
Fˆ (x ∂x ) xn = F (n) xn ,
(26)
we get I(x) =
∞ X
α(k) F (m k + p) x m k+p .
k=0
3
(27)
2
To clarify the content of this result, we consider the case g(t) = e−t , f (x) = Jn (x) (n 6= 0). By using eq. (26) one has Z ∞ ∞ x 2 k+n √ X (−1)k 1 2 √ dt Jn (x e−t ) = π . (28) k! (k + n)! 2 k + n 2 −∞ k=0
The above series has been proven to be convergent and the correctness of the proposed procedure has been checked numerically. As a further example we consider the evaluation of the integral (21) 2 for g(t) = (1 + t2 )−1 , f (x) = x2 e −x . Since Z ∞ √ Γ(a − 21 ) 1 = (29) dt π (1 + t2 )a Γ(a) −∞
we get
Z
( 2 ) ∞ √ X 1 (−1)k Γ(2 k + 32 ) 2 k x dt π x . exp − = (1 + t2 )2 1 + t2 k! Γ(2 k + 2) −∞ ∞
(30)
k=0
Let us now consider the following Laplace-type transform integral Z ∞ L(x) = dt e−t g(x t)
(31)
0
The use of eq. (22) leads to the operatorial identity
L(x) = Γ(x ∂x + 1) g(x)
(32)
from which, under the assumptions that L(x) can be expanded as in eq. (1), as a consequence of eq. (26) one obtains ∞ X ϕ(k) (−x)k . (33) g(x) = (k!)2 k=0
Furthermore, we find that the inverse of the transform (31) for the Hermite polynomials in eq. (16) is given by [n/m]
˜ (m) (x, y) = H n while the inverse transform of em (x) = (1 + e˜m (x) =
X
k=0 m x )−1
x n−m k y k , k! [(n − m k)!]2
(34)
is (|x| < 1)
∞ m−1 X (−1)k m k 1 X x = exp {ρ(n, m) x} (m k)! m n
(35)
k=0
where
2n + 1 π ρ(n, m) = exp i m
.
As a final example, let us consider the integral Z 1 du u α−1 (1 − u) β−1 f (u x) Iα,β (x) =
(Re α, Re β > 0) .
0
By using eq. (22), this integral can be written in term of the beta function as follow Iα,β (x) = B(α + x ∂x , β) f (x) 4
(36)
i.e., in terms of the gamma function Iα,β (x) =
Γ(β) Γ(α + x ∂x ) f (x) . Γ(α + β + x ∂x )
(37)
If the function f (x) satisfy eq. (1), one has ∞ X Ψ(n)
(−x)n
(38)
Ψ(n) = B(α + n, β) ϕ(n) .
(39)
Iα,β (x) =
n=0
n!
where
In this paper we have provided a few elements proving the usefulness of formal procedures to get results encompassing various aspects of operators and of special functions. Other examples could be discussed to corroborate the validity of the method. We believe, however, that the main plot of the whole technique can be quite well understood by the examples discussed here. In closing, we remark that the results we have obtained are at the level of well sounded conjectures. The relevant rigorous proof of them is out of the scope of the present article.
Acknowledgments One of us (G. D.) recognizes the warm hospitality and the financial support of the University Paris XIII, whose stimulating atmosphere provided the necessary conditions for the elaboration of the ideas leading to this paper.
References [1] B. Berndt, Ramanujan’s Notebooks, Part I, p. 295, Springer-Verlag, New York (1985); G. H. Hardy, Ramanujan. Twelve lectures on subjects suggested by his life and work, Cambridge University Press, Cambridge (1940). [2] T. Amdeberhan, V. H. Moll, Ramanujan J. 18, 91 (2009); T. Amdeberhan, O. Espinosa, I. Gonz´ alez, M. Harrison, V. H. Moll, A. Straub, Ramanujan’s Master Theorem, submitted for publication (2011). [3] D. Babusci, G. Dattoli, arXiv:1103.3947v1[math-ph]. [4] K. G` orska, D. Babusci, G. Dattoli, G. H. E. Duchamp, K. A. Penson, arXiv:1104.3406v1[math-ph]. [5] S. Roman, The Umbral Calculus, Dover Publications, New York (2005). [6] I . S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed., A. Jeffrey ed., Academic Press, New York (1994). [7] A. P. Prudnikov, Yu. A. Brychkov, and I. O. Marichev, Integrals and Series. Special Functions, vol. 2, Gordon and Breach Science Publishers, New York (1992). [8] G. E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge (2001). [9] F. G. Tricomi, Funzioni ipergeometriche confluenti, Cremonese, Roma (1954); P. Appell, J. Kamp´e de F´eri´et, Fonctions Hyperg´eometriqu´es Polynˆ ome d’Hermite, Gauthier-Villars, Paris (1926). [10] G. Dattoli, P. L. Ottaviani, A. Torre, and L. V´ azquez, Riv. Nuovo Cimento 20, n. 2 (1997). 5
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