Commun. Math. Stat. (2014) 2:413–429DOI 10.1007/s40304-015-0048-z

Krätzel Function and Related Statistical Distributions

T. Princy

Received: 8 December 2014 / Revised: 27 January 2015 / Accepted: 11 February 2015 /Published online: 17 March 2015© School of Mathematical Sciences, University of Science and Technology of China and Springer-VerlagBerlin Heidelberg 2015

Abstract The Krätzel function has many applications in applied analysis, so thisfunction is used as a base to create a density function which will be called the Krätzeldensity. This density is applicable in chemical physics, Hartree–Fock energy, heliumisoelectric series, statistical mechanics, nuclear energy generation, etc., and also con-nected to Bessel functions. The main properties of this new family are studied, showingin particular that it may be generated via mixtures of gamma random variables. Somebasic statistical quantities associated with this density function such as moments,Mellin transform, and Laplace transform are obtained. Connection of Krätzel dis-tribution to reaction rate probability integral in physics, inverse Gaussian density instochastic processes, Tsallis statistics and superstatistics in non-extensive statisticalmechanics, Mellin convolutions of products and ratios thereby to fractional integrals,synthetic aperture radar, and other areas are pointed out in this article. Finally, weextend the Krätzel density using the pathway model of Mathai, and some applicationsare also discussed. The new probability model is fitted to solar radiation data.

Keywords Laplace transform · Krätzel function · Mellin transform · H-function ·Krätzel density · Pathway model

Mathematics Subject Classification (2010) 33C60 · 44A10 · 44A15 · 60E05

T. Princy (B)Centre for Mathematical Sciences, Arunapuram P.O., Pala 686 574, Kerala, Indiae-mail: princycms@gmail.comURL: http://www.cmsintl.org

T. PrincyDepartment of Statistics, Banaras Hindu University (B.H.U), Varanasi 221005, India

123

414 T. Princy

1 Introduction

The Krätzel function was first introduced by Krätzel [9] as the kernel of an integraltransform. Properties of Krätzel function were investigated by many authors. We notethat the Krätzel function occurs in the study of astrophysical thermonuclear functions,which are derived on the basis of Boltzmann–Gibbs statistical mechanics [16]. Thegeneralized Krätzel function was examined by Kilbas and Kumar [8]. Such investiga-tions are of great interest in connection with applications. Due to the importance ofthe Krätzel function, we use this function as a base to create a density function.

This density will be shown to be connected to the following problems: Reaction rateprobability integral and its generalizations in reaction rate theory in physics, Tsallisstatistics and superstatistics in non-extensive statistical mechanics, Mellin convolu-tions of products and ratios in applied analysis, Krätzel transform, inverse Gaussiandensity in stochastic processes, Hartree–Fock energy theory, helium isoelectric series,microwave sea clutter, synthetic aperture radar, K-distribution model, neural networks,solar radiation statistics, etc.

The Krätzel function Zνρ(x) is defined for x > 0, ρ ∈ R and ν ∈ C as

Zνρ(x) =

∫ ∞

0yν−1e−yρ− x

y dy. (1.1)

In particular, when ρ = 1 and x = y2

4 , then it gives

Zν1

(y2

4

)= 2

( y

2

)ν

Kν(y),

where Kν(y) is the modified Bessel function of the third kind or McDonald functionas in [4]. We know that for all u > 0 and ν ∈ C

Zν1

(u2

4

)= 2

(u

2

)ν

Kν(u) = 2(u

2

)ν

K−ν(u),

and consequently

Zν1 (u) = 2u

ν2 Kν

(2√

u).

Note that this function is useful in chemical physics. More precisely, the function

u �→ 2ν−1 Zν1

(uq2/4

)= (

q√

u)ν

Kν

(q√

u)

is related to Hartree–Fock energy and is used as a basis function for the helium iso-electronic series, see [2,3] for more details. Moreover, the function

u �→√

2

π2ν−1 Zν

1

(u2

4

)=

√2

πuν Kν(u)

123

Krätzel Function 415

is called in the literature as reduced Bessel function and plays an important role intheoretical chemistry, see [18] and the references therein for more details.

This paper is organized as follows: First, we introduce a new class of continu-ous statistical distributions, which we name, Krätzel distributions. Some theoreticalproperties of the Krätzel density function are obtained in Sect. 2. In Sect. 3, the distrib-utions of products and ratios of independent Krätzel random variables are mentioned.In Sect. 4, we extend the Krätzel density using the pathway model of Mathai andsome applications are also discussed. In Sect. 5, we discuss an application of one ofthe above-mentioned probability models, which is shown by fitting the model to solarradiation data.

2 A Density Function

In this section, we introduce the Krätzel distribution and study the basic distributionalproperties.

A random variable X is said to have a Krätzel distribution with parameters α, ν andρ if its density function is given by

f (x) ={

c xα−1 Zνρ(x), x ≥ 0, ρ > 0, α > 0, ν > 0,

0, otherwise,(2.1)

where

Zνρ(x) =

∫ ∞

0yν−1e−yρ− x

y dy

and c is the normalizing constant, which can be evaluated using a gamma integral andit is given by c = ρ

�(α) �(

α+νρ

) . A family of densities is available for various values of

the parameters. The graphs of Krätzel density for various values of the parameters aregiven in the following figures.

Figure 1 illustrates the Krätzel density when the parameters ρ and ν are fixed andα is changing. Note that the parameter α is called the shape parameter in the Krätzelcase because α throws light on the shape of the density curve. Figure 2 shows thedensity when α and ρ are fixed and ν is varying.

The Mellin transform technique is used to determine the moments and some prop-erties of the proposed distribution.

2.1 Mellin Transform

The Mellin transform of the density function f (x) is the (s −1)th moment of a positiverandom variable, where s is a complex parameter. The Mellin transform is given by

M f (s) = E(Xs−1)

= 1

�(α)�(

α+νρ

)�(α + s − 1)�

(α + ν + s − 1

ρ

), (2.2)

123

416 T. Princy

Fig. 1 Krätzel density functions: ρ = 3 and ν = 3 fixed, α changing

Fig. 2 Krätzel density functions: ρ = 3 and α = 4 fixed, ν changing

123

Krätzel Function 417

for �(α + s − 1) > 0, ν > 0, ρ > 0. From (2.2) we may also observe the followingproperties. If U is the Krätzel random variable, then we have

E(U s−1) = 1

�(α)�(

α+νρ

)�(α + s − 1)�

(α + ν + s − 1

ρ

)

= E(Y s−1)E(Zs−1), (2.3)

where

E(Y s−1) = �(α + s − 1)

�(α)for �(α + s − 1) > 0 (2.4)

and

E(Zs−1) =�(

α+ν+s−1ρ

)

�(

α+νρ

) for �(α + ν + s − 1) > 0, ρ > 0. (2.5)

This suggests that the Krätzel random variable has the representation U = Y Z , whereY is a gamma random variable with parameter (α, 1), Z is a generalized gammarandom variable, U is a Krätzel random variable, and it is assumed that Y and Z areindependently distributed. This will be stated as a theorem.

Theorem 2.1 The Krätzel random variable X of (2.1) has the structural representa-tion X = Y Z , where Y and Z are independently distributed with Y having a gammadensity with parameters (α, 1) and Z having a generalized gamma density with para-meters (α, ν, ρ).

Using the Mellin transform of the density function, we give a representation of theKrätzel density in terms of H -function. Hence all the properties of H -function cannow be made use of to study this density function further. The density function f (x)

of (2.1) in terms of H -function is given by

f (x) = 1

�(α)�(

α+νρ

) H2,00,2

⎡⎣x

∣∣∣∣(α−1,1),

(α+ν−1

ρ, 1ρ

)

⎤⎦ , x ≥ 0, α > 0, ρ > 0, ν > 0

(2.6)and f (x) = 0 elsewhere. Let us consider the special case for ρ = 1, then f (x) canbe written as

f (x) = 2

�(α)�(α + ν)xα+ ν

2 −1 Kν(2x12 ), x ≥ 0, α > 0, ν > 0 (2.7)

and f (x) = 0 elsewhere, where Kν(2x12 ) is the modified Bessel function of the third

kind or McDonald function. When ρ is real and rational, the H -function appearing in(2.6) can be reduced to G-function using the multiplication formula for the gammafunction, namely,

�(mz) = (2π)1−m

2 mmz− 12 �(z)�

(z + 1

m

). . . �

(z + m − 1

m

), m = 1, 2, . . . .

123

418 T. Princy

As a consequence, when ρ = 1m we have

f (x) = 1

mm�(α)�(α + ν)�(α + ν + 1m ) . . . �(α + ν + 1 − 1

m )

× Gm+1,00,m+1

[x

mm

∣∣∣∣α−1,α+ν−1,α+ν+ 1

m −1,...,α+ν− 1m

], (2.8)

x ≥ 0, α > 0, ν > 0, m = 1, 2, . . . and f (x) = 0 elsewhere. When m = 2, f (x)

becomes

f (x) = 1

4�(α)�(α + ν)�(α + ν + 12 )

G3,00,3

[x

4

∣∣∣∣α−1,α+ν−1,α+ν+ 1

2

], (2.9)

x ≥ 0, α > 0, ν > 0. This is a real physical situation (reaction rate theory), see forexample Mathai and Haubold [12].

2.1.1 Remark

The representation (2.6) of the Krätzel density function f (x) as a Fox’s H2,00,2 -function

reveals it as a special case (m = 2) of the Kernel function Hm,00,m , m > 1. When α = 1, it

is a very general H -integral transform generalizing the Laplace and Meijer transforms.For ρ = 1, the Krätzel function as mentioned before is the modified Bessel functionof the third kind (McDonald function) representable as a Meijer’s G2,0

0,2-function.From (2.2), the mean and variance of the Krätzel density function can be derived

as follows:

E(X) = α�(

α+ν+1ρ

)

�(

α+νρ

) , ρ > 0, α + ν > 0, (2.10)

Var(X) = α(α + 1)�(

α+ν+2ρ

)

�(

α+νρ

) −⎛⎝α

�(

α+ν+1ρ

)

�(

α+νρ

)⎞⎠

2

, α + ν > 0, ρ > 0.

(2.11)

Theorem 2.2 Let Y ≥ 0 be a gamma random variable with parameters (α, 1) andZ ≥ 0 be a generalized gamma random variable with parameters (α + ν, 1, ρ) whereρ is the power and α + ν is the shape parameter, and let Y and Z be independentlydistributed. Then U = Y Z is a Krätzel random variable with the Laplace transform

L f (t) = 1

�(α) �(

α+νρ

) H1,22,1

[t

∣∣∣∣(1−α,1),(1− α+ν

ρ, 1ρ)

(0,1)

], t > 0. (2.12)

123

Krätzel Function 419

Proof The Laplace transform L f (t) of the density function (2.1) is given by

L f (t) =∫ ∞

0e−t x f (x) dx . (2.13)

Using (2.1) and (2.13) and changing the order of integration, we have

L f (t) = ρ

�(α) �(

α+νρ

)∫ ∞

0uν−1e−uρ

{∫ ∞

0xα−1 e−t x− x

u dx

}du.

Making the substitution v = xu and using the gamma integral, we get

L f (t) = ρ

�(

α+νρ

)∫ ∞

0uα+ν−1 e−uρ

(1 + ut)−α du, (1 + ut) > 0. (2.14)

We can evaluate this integral using Mellin convolution properties, for details seeAppendix. Then the Laplace transform of the density function f (x) can be obtainedin terms of H -function and it is given by

L f (t) = 1

�(α) �(

α+νρ

) H1,22,1

[t

∣∣∣∣(1−α,1),(1− α+ν

ρ, 1ρ)

(0,1)

], t > 0. (2.15)

2.2 Some Properties of the Krätzel Function

Log-concavity and log-convexity of functions and sequences in probability have beenof interest to several authors, see for example Karlin [7]. Log-convexity is of interestin the study reliability and infinitely divisible random variables. In this section, wepresent log-convexity and complete monotonicity of the Krätzel density.

A function φ on (0,∞) is said to be completely monotone if φ has derivatives ofall orders and satisfies

(−1)nφ(n)(λ) ≥ 0

for all λ > 0 and n ∈ {0, 1, ...}, and as λ → 0 the value of φ(n)(λ) approaches a finitelimit which we denote by φ(n)(0). A function g on (0,∞) is said to be logarithmicallyconvex, or simply log-convex, if its natural logarithm ln g is convex, that is, for allx1, x2 > 0 and α ∈ [0, 1], we have

g(αx1 + (1 − α)x2) ≤ [g(x1)]α[g(x2)]1−α.

Theorem 2.3 If ρ, ν, α ∈ (0,∞) and x ≥ 0, then the following assertions are true:

a. The Krätzel density f (x;α, ν, ρ) satisfies the recurrence relation

ν f (x;α, ν, ρ) = (α + ν) f (x;α, ρ + ν, ρ) − α f (x;α + 1, ν − 1, ρ). (2.16)

123

420 T. Princy

b. The density function x �→ f (x) is completely monotonic on (0,∞) if 0 < α ≤ 1.c. The function x �→ f (x) is log-convex on (0,∞) if 0 < α ≤ 1.

Proof (a) Using (2.1) and the recurrence relation of the Krätzel function [1],

νZνρ(x) = ρZν+ρ

ρ (x) − x Zν−1ρ (x). (2.17)

Now the rest of the derivation in this section is parallel to those in Princy [15]. Forexample, consider

ν f (x;α, ν, ρ) = νρ

�(α) �(

α+νρ

) xα−1 Zνρ(x)

= ρxα−1

�(α) �(

α+νρ

) [ρZν+ρρ (x) − x Zν−1

ρ (x)]

= ρ2

�(α) �(

α+νρ

) xα−1 Zν+ρρ (x) − ρ

�(α) �(

α+νρ

) xα Zν−1ρ (x)

=ρ2

(α+νρ

)

�(α)(

α+νρ

)�(

α+νρ

) xα−1 Zν+ρρ (x)− ρα

α�(α) �(

α+νρ

) xα Zν−1ρ (x)

= (α + ν)ρ

�(α) �(

α+νρ

+ 1) xα−1 Zν+ρ

ρ (x)

−αρ

�(α + 1) �(

α+νρ

) xα Zν−1ρ (x)

= (α + ν) f (x;α, ν + ρ, ρ) − α f (x;α + 1, ν − 1, ρ).

This implies that

ν f (x;α, ν, ρ) = (α + ν) f (x;α, ν + ρ, ρ) − α f (x;α + 1, ν − 1, ρ).

(b) Letf (x) = g1(x) g2(x), (2.18)

where g1(x) = cxα−1, g2(x) = Zνρ(x).

The function g1(x) is obviously completely monotonic because c is normalizingconstant and xα−1 is completely monotonic on 0 < α ≤ 1.

Next we consider

Zνρ(x) =

∫ ∞

0tν−1e−tρ− x

t dt. (2.19)

The change of variable 1t = s in (2.19) yields

Zνρ(x) =

∫ ∞

0(s−ν−1e−s−ρ

) e−xsds.

123

Krätzel Function 421

That is, the Krätzel function Zνρ(x) is the Laplace transform of the function s �→

s−ν−1e−sρ. Thus in the view of Bernstein–Widder theorem which implies that the

function x �→ Zνρ(x) is completely monotonic, i.e., for all n ∈ {0, 1, 2, ...}, ν, ρ ∈

(0,∞) and x > 0, we have (−1)n[Zνρ(x)](n) > 0. This proves that g2(x) is completely

monotonic.Next we use the Leibnitz formula for the nth derivative of a product, we have

(−1)n f (n)(x) ≥ 0. Therefore x �→ f (x) is completely monotonic.(c) Every completely monotonic functions is log-convex, see [[19], p. 167].

2.3 Particular Cases of Interest

Some of the product distributions are special cases of Krätzel distribution as intro-duced in Sect. 1. The distributions of products of random variables are of great interestin many areas of science, especially in biology, physical sciences and econometrics.In traditional portfolio selection models, certain cases involve the product of randomvariables. The best example of this is in the case of investment in a number of differ-ent overseas markets. Some of the product distributions are special cases of Krätzeldistribution. For instance, the following distributions arise as a particular case of (2.1):

1. When X and Y are independently distributed random variables having the gammaand Weibull distributions, respectively, then the density function of XY is obtainedfrom (2.1) by letting α + ν = ρ. The product of gamma and Weibull randomvariables is studied in detail by Nadarajah and Kotz [14].

2. When X and Y are independently distributed random variables having the gammaand Rayleigh distributions, respectively, then the density function of XY isobtained from (2.1) by letting α + ν = 2 and ρ = 2. The product of gammaand Rayeligh random variables is studied in detail by Shakil and Kibria [17].

3. When X and Y are independently distributed random variables having one parame-ter gamma distributions, then the density function of XY is obtained from (2.1) byletting ρ = 1. The product of two gamma variables is studied in detail by LorenzoZaninetti [20]. This density function is used to describe the mass distribution ofgalaxies.

3 Evaluation of Densities of Products and Ratios

Theorem 3.1 Let Xi , i = 1, 2 be two independently distributed random variableshaving Krätzel densities

f (xi ) ={

ci xαi −1i Zνi

ρi (xi ), xi ≥ 0, ρi > 0, αi > 0, νi > 0,

0, otherwise,(3.1)

where

Zνiρi

(xi ) =∫ ∞

0tνi −1 e−tρi − xi

t dt

123

422 T. Princy

and ci = ρi

�(αi ) �(αi +νi

ρi). Then the density of Y = X1

X2is given by

g(y) =⎧⎨⎩

2∏i=1

1

�(αi ) �(

αi +νiρi

)⎫⎬⎭ H2,2

2,2

⎡⎣y

∣∣∣∣(−α2,1),(1− α2+ν2+1

ρ2, 1ρ2

)

(α1−1,1),(

α1+ν1−1ρ1

, 1ρ1

)

⎤⎦ , y ≥ 0, (3.2)

and the density of Z = X1 X2 is given by

h(z)=⎧⎨⎩

2∏i=1

1

�(αi ) �(

αi+νiρi

)⎫⎬⎭ H4,0

0,4

⎡⎣z

∣∣∣∣(αi −1,1),

(αi +νi −1

ρi, 1ρi

), i=1,2

⎤⎦ , z ≥ 0. (3.3)

Proof Here we evaluate the density of Y = X1X2

by the method of Mellin transform,

E(Y s−1) = E(Xs−11 ) E(X−(s−1)

2 )

=�(α1 + s − 1) �(α2 − s + 1) �

(α1+ν1+s−1

ρ1

)�(

α2+ν2−s+1ρ2

)

�(α1) �(α2)�(

α1+ν1ρ1

)�(

α2+ν2ρ2

) ,

(3.4)

�(αi ) > 0, �(α1 + s − 1) > 0, �(α2 − s + 1) > 0,�(α1 + ν1 + s − 1) > 0,

�(α2 + ν2 − s + 1) > 0, ρi > 0, i = 1, 2. By taking the inverse Mellin transform,we will get the density of Y = X1

X2as in (3.2). Similarly, we can establish the density

of Z = X1 X2. Note that for ρi = 1, i = 1, 2, is of the form ρi = 1mi

, mi = 1, 2, . . . ,

then in (3.2) and (3.3) reduce to Meijer’s G-functions G2,22,2 and G4,0

0,4 respectively.

4 Extension of the Krätzel Density Using Pathway Model

The scalar version of the pathway model of Mathai [13] is shown to be associatedwith a large number of probability models used in physics and statistics. The originalpathway model of Mathai [13] is for the rectangular matrix case. The scalar versionof the pathway model is as follows:

f1(x) = c1|x |γ [1 − a(1 − α)|x |δ] η1−α , − ∞ < x < ∞ (4.1)

for a > 0, η > 0, δ > 0, γ > −1 and c1 is a constant. A statistical density can becreated out of f1(x) under the additional conditions 1 − a(1 −α)|x |δ > 0 and then c1will act as a normalizing constant. Hence the model is applicable to random or non-random situations as well. For α < 1, the model in (4.1) will stay in the generalizedextended type-I beta family of functions. For α > 1, writing 1 − α = −(α − 1) themodel switches into the form

f2(x) = c2|x |γ [1 + a(α − 1)|x |δ]− ηα−1 (4.2)

123

Krätzel Function 423

for α > 1,−∞ < x < ∞, a > 0, η > 0, δ > 0 which is a generalized extendedtype-II beta family of functions. Again taking c2 as the normalizing constant, onecan create a statistical density in f2(x). When α goes to 1 from the right or from theleft, f1(x) and f2(x) will go into a generalized extended gamma family of functions,namely

f3(x) = c3|x |γ e−aη|x |δ , (4.3)

where a > 0, η > 0, δ > 0. The model in (27)covers almost all statistical densitiesin current use in statistics, physics, engineering and other areas. Here we extend theKrätzel density using the pathway model.

An extended form of the Krätzel density is given by

f4(x) ={

c4xα−1 Zν,βρ,δ (x), x ≥ 0, α > 0, ν > 0, ρ > 0,

0, otherwise,(4.4)

where

Zν,βρ,δ (x) =

∫ ∞

0tν−1[1 + (β − 1)tρ]− 1

β−1 e− xtδ dt (4.5)

and c4 is the normalizing constant.More families are available when the variable is allowed to vary over the real line.

So f4(x) is a more general class of the density function. When α = 1, the densityfunction f4(x) becomes

f5(x) ={

c5 Zν,βρ,δ (x), x ≥ 0, ν > 0, ρ > 0,

0, otherwise,(4.6)

where Zν,βρ,δ (x) is given in (4.5).

The graph of the extended form of the Krätzel density for various values of para-meters are given in the following figures (Fig. 3).

The integral involved in the density function (4.6) is connected to nuclear reac-tion rate theory in the non-resonant case, Tsallis statistics in non-extensive statisticalmechanics, superstatistics in astrophysics, generalized type-1 and type-2 beta, gammafamilies of densities, density of a product of two real positive random variables, Krätzelintegral in applied analysis, inverse Gaussian density in stochastic processes, etc. Spe-cial cases include a wide range of functions appearing in different disciplines.

When α = 1 and β → 1, the density function f4(x) becomes

f6(x) ={

c6 Zν,1ρ,δ(x), x ≥ 0, ν > 0, ρ > 0,

0, otherwise,(4.7)

where

Zν,1ρ,δ(x) =

∫ ∞

0tν−1 e−tρ− x

tδ dt. (4.8)

123

424 T. Princy

Fig. 3 The graph of the extended form of the Krätzel density for various values of parameters

When δ = −η, η > 0 in f6(x) the integral involved in the density function is denotedby I2 and it is given by

I2 =∫ ∞

0tν−1 e−tρ−xtηdt. (4.9)

This integral can be evaluated using the convolution properties of Mellin transform,we get

I2 =∫ ∞

0tν−1 e−tρ−xtη dt = 1

ρηH1,1

1,1

[x

1η

∣∣∣∣(1− ν

ρ, 1ρ)

(0, 1η)

], x > 0. (4.10)

From this, we can make connection to Laplace transform of some related densities.Some of them are listed below:

• For η = 1 in (4.9), I2 becomes the Laplace transform of the generalized gammadensity with Laplace parameter x . When x is replaced by −x , we have the momentgenerating function there. Similar remarks apply to the following cases also.

• For η = 1, ν = ρ in (4.9), I2 becomes the Laplace transform of Weibull density.• For η = 1, ρ = 1 in (4.9), I2 becomes the Laplace transform of the gamma density.• For η = 1, ν = 3, ρ = 2 in (4.9), I2 becomes the Laplace transform of Maxwell–

Boltzmann density.

123

Krätzel Function 425

4.1 Some Applications

• For α = 1, f (x) gives the Krätzel integral

f (x) =∫ ∞

0tν−1e−tρ− x

t dt, x ≥ 0, (4.11)

which was studied in detail by Krätzel [9]. An additional property is that it can bewritten as H -function of the type H2,0

0,2 (.).• Over the last decade, significant progress has been made toward the development

of widely applicable radar clutter models. The K-distribution has proved to be anappropriate model for characterizing the amplitude of microwave sea clutter [6]. TheK-distribution is a model for the statistics of synthetic aperture radar imagery. The K-distribution that appears in different areas such as radiation, ultrasonic backscatter,neural networks, etc was studied by many authors. One form of the K-distributionis as follows:

g1(u) = 2

�(ν + 1)

(u

2

)ν

Kν(u), u ≥ 0. (4.12)

For ρ = 1, α = 1 and x = u2

4 in (2.1), the Krätzel density function coincides withthe K-distribution. Hence we can say that Krätzel distribution is a new generalizationof the K-distribution.

• In a series of papers, Haubold and Mathai studied modifications of Maxwell Boltz-mann theory of reaction rates. The basic reaction rate probability integral has thefollowing form

I3 =∫ ∞

0tγ−1e−at−zt−

12 dt, (4.13)

which is the non-resonant case of nuclear reactions. Compare integral (4.13) with(4.7). The reaction rate probability integral (4.13) is (4.7) for ρ = 1, δ = 1

2 , ν = γ

and x = z. The basic integral (4.13) is generalized in many different forms forvarious situations of resonant and non-resonant cases of reactions, depletion ofhigh energy tail, cut-off of the high energy tail, and so on, see for example Mathaiand Haubold [11,12], Haubold and Kumar [5].

5 Application in Solar Radiation Data

The probability model in (2.1) has application in the solar modeling. In the followingsection, we discuss how well this model describes the solar radiation data.

5.1 Data Analysis

Data were collected from NREL-National Renewable Energy Laboratory, U.S. Depart-ment Energy. The data consist of wavelength and the corresponding solar radiance andthey consist of 2000 observations. Here, mathematical software MAPLE and MAT-LAB are used for the data analysis. The summary of the statistics obtained from thedata is as follows:

123

426 T. Princy

Fig

.4T

hegr

aph

ofth

epr

obab

ility

mod

elfit

ted

toth

eso

lar

radi

atio

nda

ta

123

Krätzel Function 427

Mean value = 624.7485254,

Variance = 746556.501.

Fixing ρ = 1 and using the method of moments, the estimates of the parameters inthe Krätzel density function are obtained as follows:

α̂ = 5.1718,

ν̂ = 1.1955.

Figure 3 shows the histogram of the data embedded within the probability model.Using χ2-test, we obtained the calculated χ2 value as 5.451 and the tabulated value

as 11.07. Hence, we conclude that the model is consistent for the data (Fig. 4).

Acknowledgments The author would like to thank Professor A. M. Mathai of the Centre for MathematicalSciences and Professor S. K. Singh of Banaras Hindu University for fruitful discussions. The author wouldlike to thank the University Grants Commission, Government of India, New Delhi, for the financial assistancefor this work under Rajiv Gandhi National Fellowship, and the Centre for Mathematical Sciences forproviding all facilities.

Appendix

H -function is defined as the following Mellin–Barnes integral:

Hm,np,q

[z

∣∣∣∣(a1,α1),...,(ap,αp)

(b1,β1),...,(bq ,βq )

]= 1

2π i

∫L

φ(s)z−s ds, i = √−1, (6.1)

where

φ(s) = {∏mj=1 �(b j + β j s)}{∏n

j=1 �(1 − a j − α j s)}{∏q

j=m+1 �(1 − b j − β j s)}{∏pj=n+1 �(a j + α j s)}

, (6.2)

where a j , j = 1, 2, . . . , p and b j , j = 1, 2, . . . , q are complex numbers,α j , j = 1, 2, . . . , p and β j , j = 1, 2, . . . , q are positive real numbers, and Lis a contour separating the poles of �(b j + β j ), j = 1, 2, . . . , m from those of�(1 − a j − α j s), j = 1, 2, . . . , n. Convergence conditions, properties and applica-tions of H -function in various disciplines are available in the literature. Computablerepresentations and series forms are also available. For example, see [10]. Whenα j = 1, j = 1, 2, . . . , p and β j = 1, j = 1, 2, . . . , q, the H-function in (6.1)

reduces to Meijer’s function Gm,np,q

[z|a1,...,ap

b1,...,bq

].

Evaluation of (2.14). Let

I1 =∫ ∞

0uα+ν−1 e−uρ

(1 + ut)−α du. (6.3)

We can evaluate this integral using Mellin convolution properties, by taking it as astatistical distribution problem for the sake or novelty. Let X1 and X2 be independentlydistributed real scalar positive random variables. Let the densities be h1(x1) and h2(x2),

123

428 T. Princy

respectively. Consider the transformation W = X1X2

and V = X2. Then the density ofW is denoted by

g(w) =∫

v

vh1(wv)h2(v) dv. (6.4)

Also,E(W s−1) = E(Xs−1

1 ) E(X1−s2 ) (6.5)

due to independence of X1 and X2. But when h1(x1) and h2(x2) are densities andwhen X1 and X2 are real positive variables, then E(Xs−1

1 ) = Mh1(s) and E(X1−s2 ) =

Mh2(2 − s). Let

h1(x1) ={

c1 (1 + x1)−α, x1 ≥ 0, α > 0,

0, otherwise,(6.6)

and

h2(x2) ={

c2 x2α+ν−2 e−xρ

2 , x2 ≥ 0, ρ > 0, α > 0, ν > 0,

0, otherwise,(6.7)

where c1 and c2 are the normalizing constants. Then the density of W = X1X2

is givenby

g(w) = c1c2

∫ ∞

0v vα+ν−2 e−vρ

(1 + wv)−α dv. (6.8)

Comparing (6.3) and (6.8), we see that the integral I1 is available from (6.8) for w = t .Now let us evaluate the density using Mellin transform. Therefore the density of u isgiven by

g(w) = 1

2π i

∫ c+i∞

c−i∞Mh1(s)Mh2(2 − s)w−s ds. (6.9)

But

E(Xs−11 ) = c1

�(s)�(α − s)

�(α), for �(s) > 0,�(α − s) > 0, (6.10)

E(X1−s2 ) = c2

ρ�

(α + ν − s

ρ

), for �(α + ν − s) > 0, ρ > 0. (6.11)

Hence from (6.9)–(6.11),

∫ ∞

0v vα+ν−2e−vρ

(1 + wv)−α dv = 1

ρ�(α)H1,2

2,1

[w

∣∣∣∣(1−α,1),(1− α+ν

ρ, 1ρ)

(0,1)

]. (6.12)

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