Some definite integrals arising from selfdecomposable characteristic functions

In the probability theory, selfdecomposable or class L0distributions play an important role as they are limit distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class L0 is quite large and includes many known classical distributions. For this note, the most important feature of the selfdecomposable variables are their random integral representation with respect to a Lévy process. From those random integral representations we get the equality of logarithms of some characteristic functions. These allow us to get formulas for some definite integrals; some of them were previously unknown, and some are rarely quoted in popular tables of integrals and series.

The class L of selfdecomposable distributions appears in the classical probability theory as a class of limiting distributions obtained from infinitesimal triangular arrays arising from sequences of independent random variables.In the case of independent and identically distributed variables we get the class S of stable distributions, which contains the normal distributions.On the other hand, when one considers arbitrary infinitesimal triangular arrays we end-up with the class ID of infinitely divisible distributions.Thus we have the inclusions S L ID.
The measure M is called the Lévy measure of X.The above integral formula is refered to as the Lévy-Khintchine representation.The triple a, σ 2 and M is uniquely determined by X.The classic references to those topics are: Feller (1966), Gnedenko and Kolomogorov (1954) or Loeve (1963).
For infinite divisibility for Banach space valued variables we refer to Araujo and Gine (1980), Chapter 3.
The aim of this note, (after the preliminary Proposition 1, in the Introduction), is to show how some definite integrals can be computed from the selfdecomposable characteristic functions; cf.Lemmas 1 , 2 and Corollary 1 (for the hyperbolic characteristic functions) and Corollaries 2-6, in Sections 2 and 3 (for the characteristic function expressed via Euler's Euler's gamma function; in particular, for the Meixner or Feller-Spitzer distributions).

Introduction.
A random variable X is called selfdecomoposable(in symbols: X ∈ L 0 ), if (1) And inductively, for k ≥ 1, we define, decreasing sequence of the Urbanik classes L k , as follows: 1972),(1973); (1a) The "remainders" (X t , t > 0) in (1) satisfy the following cocycle equation X t+s d = e −t X s + X t .It allows to construct a cadlag Lévy process and to infer that the selfdecomposability of X is equivalent to the claim that there exits unique, in distribution, a Lévy process (Y X (t), t ≥ 0), such that Jurek and Vervaat (1983),Theorem 3.2, pp. 252-253 or Jurek and Mason (1993), Theorem 3.6.8, pp. 124-126 and Jurek (1982) for a Banach space valued random variables.
To the random variable Y X (1) we refer to as the background driving random variable of X; is short: BDRV.
The above equality (5) is the key identity for all the definite integrals in this note.
(ii) From (4) we see that X ∈ L 0 has Gaussian part if and only if its BDRV Y X (1) has a Gaussian part.
(iii) For the real characteristic functions, as in Section 1 below, we get simpler formulas as we can discard the imaginary part.

The hyperbolic characteristic function.
For the information about the hyperbolic-type characteristic function we refer to Pitaman and Yor (2003) or Jurek and Yor (2004).Also this section may be viewed as a complement to the recent preprint Jurek (2022 a).a).Hyperbolic-sine function.
Lemma 1. From the selfdecomposability of the hyperbolic-sine characteristic function φ Ŝ (t) = t sinh(t) we get the following formulas (Lévy exponents): i.e., in this example in a = 0 and σ 2 = 0 for the random variable X := Ŝ.
as in view of symmetry of h Ŝ (x) we have b = 0 and s 2 = 0 in (4) above.
On the other hand, as ψ we get the part (i) of Lemma 1.Since is positive and also symmetric we infer by Proposition On the other hand , which together with (5) gives the part (ii) of Lemma 1.
Next, using Wolframalpha, we have that is non-negative function therefore, by the part (c) of Proposition 1, we get that Ŝ ∈ L 2 , (Urbanik class).Finally, since from ( 5) we get the part (iii) in Lemma 1.
and that fact was used in the proof of Lemma 1.
Hence for h On the other hand, by (3), and (5) As Y Ĉ (1) ∈ L 0 (is selfdecomposable) we can repeat the procedure from the previous step.Since, by WolframAlpha, is a positive and symmetric density of a Lévy measure of BDRV of Y Ĉ (1) in (2).
respectively.By (e) in Proposition 1 we get that T ∈ L 0 .Then its BDRV Y T (1) = [0, 0, h T (x)], where Hence we get Corollary 1. (i) From the selfdecomposability of hyperbolic tangent T we get equality (ii) The characteristic function ψ T (t) represents a compound Poisson distribution therefore T / ∈ L 1 Urbanik class.
For the part (i), also cf.Jurek-Yor (2004), Proposition 1 with Corollary 1 and the equality (10).Since Lévy measures of class L 0 are infinite by (4), we get the part (ii) of corollary.
Remark 2. Formulas Lemma 1(i), Lemma 2(i) and the above had already appeared in Jurek-Yor (2004).They are added here for the completeness of this presentation.

Characteristic functions expressed via the Euler's gamma function.
(a).The log-gamma distribution.
On the other hand, from (3), we get All in all, from the identity (5) (taking the Kolmogorov's kernel) we infer the identity Corollary 2. From the selfdecomposability property of the log-gamma variables, for α > 0, β > 0 and t ∈ R we have (1)
On the other hand from (3) we get All in all we get Corollary 3. From the selfdecomposability property of the logistic distribution l α , α > 0, for t ∈ R, we have where B(z 1 , z 2 ) denotes the beta-function; cf.Schoutens (2003) , p. 64, or Ushakov (1999) , p. 309.(For a particular choices of parameters we get Fisher z-distribution; cf.Jurek (2021), the section 3.14, p.105.) Let us note that the characteristic function φ GZ (t) can be expressed via characteristic functions of log-gamma variables.Namely, as we have cf. the section (b), on log-gamma variables, above.Below we will assume a = 2π, d = 1/2 and m = 0 and denote Hence by (7) and section (a) on the log-gamma variables, the GZ distribution has Lévy (spectral) measure ν GZ (dx) := k GZ (x)dx where the density is of the from Hence, is the density of Lévy measure of the BDRV Y GZ (1).From Corollary 2, in Jurek ( 2022), we have that log-gamma variables have finite second moment.Consequently, in Kolmogorov's representation, we have On the other hand, from (5) we get that where Ψ (0) (z) := dLogΓ(z)/dz is the digamma function.Hence by (6) we infer the following: Corollary 4. From the selfdecomposability of the generalized z-distribution we have the identity for all t ∈ R. ) 2d exp(imt); cf. Schoutens (2003), p. 62-63.From above we infer that M are infinitely divisible.Furthermore, for our purposes, without the loss of generality we assume that m = 0 and d = 1/2.Being infinitely divisible, with finite second moment, Meixner distributions admit the following representations is positive as |b| < π.Hence we infer that Meixner M ∈ L 0 .On the other hand, from ( 5) All in all we get Corollary 5. From the selfdecomposability property of Meixner M(a, b, 1/2, 0) variable with constants a > 0, |b| < π we have the identity ).
Remark 3. As an addition to the above : ).
Vinogradov and Paris (2021) or Ushakov p.283; it is called there Bessel distribution.For further generalizations of F S distributions cf.Vinogradov and Paris (2021).From above we get that F S variable is in ID class and it has Lévy-Khintchine representation