An application of theory of distributions to the family of λ-generalized gamma function

Abstract: Gamma function and its generalizations always have played a basic role in various disciplines. The aim of present study is to investigate a new representation of the λ-generalized gamma function. This representation is developed by using different modified forms of delta function. This development explores their extended use as generalized functions (distributions), which are meaningful to exist over some particular space of test functions. Further to this a discussion is presented for the suitable applications of this new representation.


Distributions and test functions
Corresponding to each space of test functions there is a dual space known as space of distributions (or generalized functions). Consideration of such functions is vital due to their important property of representing the singular functions. In this way, one can apply different operations of calculus as in the case of classical functions. For the requirements of this investigation we need to mention about delta function, which is a commonly used singular function given by and An ample discussion and explanation of distributions (or generalized functions) has been presented in five different volumes by Gelfand and Shilov [37]. Functions having compact support and infinitely differentiable as well as fast decaying are commonly used test functions. The spaces containing such functions are denoted by and respectively. Obviously, corresponding duals are the spaces ′ and ′ . A mentionable fact about such spaces is that and ′ do not hold the closeness property with respect to Fourier transform but and ′ do. In this way it is remarkable that the elements of ′ have Fourier transforms that form distributions for entire functions space whose Fourier transforms belong to [38]. Further to this explanation, it is noticeable that as the entire function is nonzero for a particular range ω 1 < < ω 2 , but zero otherwise so the following inclusion of above mentioned spaces holds More specifically, space Z comprise of entire and analytic functions sustaining the subsequent criteria |s q ℘(s)| ≤ C e η|θ| ; (q ∈ ℕ 0 ).
Throughout in this paper, except if mentioned particularly the conditions for the involved parameters are taken as stated in Sections (2.1) and (2.2).

New Representation of λ-generalized gamma function
In this section, computation of λ-generalized gamma function is given as a series of complex delta function but the discussion about its rigorous use as a generalized function over a space of test functions is a part of the next section. Theorem 1. λ-generalized gamma function has the subsequent series representation Proof. A replacement of = and = + in the integral representation of λ-generalized gamma function as given in (6) yields the following Then the involved exponential function can be represented as Next, combining the expressions (19) and (20) leads to the following which gives The actions of summation and integration are exchangeable because the involved integral is uniformly convergent. An application of identity (14) produces the following A combination of these Eqs (22) and (23) yields the required result (18). □ Corollary 1 λ-generalized gamma function has the following series form Proof. Eq (24) can be obtained by considering the following combination of Eq (16) as well as Eq (23) Next, by making use of this relation in (18) leads to the required form. □ Corollary 2 λ-generalized gamma function has the following series form Proof. Eq (23) can be rewritten as follows Next, by making use of this relation in (18) leads to the required form. □ Corollary 3 λ-generalized gamma function has the following series form Proof. A suitable combination of Eqs (16) and (26) gives which is a key to the required form. Remark 1. It is to be remarked that the following results are straightforward from the above corollaries for λ = 1 Now, by putting = 0 leads to the following [24] It is noticeable that the above series representations are given in the form of delta function. Such functions make sense only if defined as distributions (generalized functions) over a space of test functions as discussed in Section (1.2). Consequently, one needs to be very careful to choose a suitable function for which this representation holds true. As an illustration, one can put = 0 in identity (26) and multiply it by 1 Γ 0 λ (s;a) to get the following Therefore, singular points of delta function at s = −n are canceled with the zeros of Γ 0 λ (s; a) in this expression i.e lim Hence, by making use of in the above statement (37), one can get the following which is false or inconsistent. At the same time, a consideration of the following special product gives the following Since 1 Γ 0 (−n;a) = 0 due to the poles of gamma function and we get Therefore, one needs to be very careful in making a choice of function to analyse the behavior of new series representation that is discussed in the next subsection.

Analysis of the behavior of new representation
λ-generalized gamma function Γ (s; a) is expressed in a new form involving singular distributions namely delta function. Therefore, it is proved in the subsequent theorem that this new form of Γ (s; a) is a generalized function (distribution) over (space of entire test functions).
Theorem 2 Prove that Γ (s; a) acts as a generalized function (distribution) over .
Proof. For each ℘ 1 (s), ℘ 2 (s)ϵ and c 1 , c 2 ϵℂ Then, for any sequence {℘ κ } κ=1 ∞ in Z converging to zero one can assume that {〈δ(s + n − λr)), ℘ κ 〉} κ=1 ∞ → 0 due to the continuity of δ(s) Henceforth, λ-generalized gamma function is a generalized function (distribution) over test function space due to the convergence of its new form (26) explored below whereas, One can observe that ∀℘ϵΖ; ℘(λr − n) are functions of slow growth as well as exists and is rapidly decreasing. Consequently, for ∀℘( ) ; 〈Γ (s; a), ℘(s)〉 as a product of the functions of slow growth and rapid decay is convergent. Similarly, other special cases as given in (30)(31)(32)(33)(34)(35)(36) are also meaningful in the sense of distributions. This fact is also obvious by making use of basic Abel theorem. □ Hence the behavior of this new series is discussed for the functions of slow growth but it is mentionable that this new series may converge for a larger class of functions. Consequently Similarly, by considering the distributional form of generalized gamma function as given in (32), we obtain the following specific form of (49) with λ = a = 1 (50) Remark 2. Sequences as well as sums of delta function have significant importance in diverse engineering problems, for example these are used as an electromotive force in electrical engineering. This is noticeable that if one multiplies {δ(s + n − r)} =0 ∞ with 2 exp(− − ) then it will produce the distributional representation of λ-generalized gamma function. Furthermore, if one takes = 1 = λ; = 0, then related outcome do hold for special cases as well. This discussion illustrates the possibility of further important identities. For instance if one considers = −1 in (49) then it will compute Laplace transform of Γ (s; a). Therefore it becomes more important to check the validation of such results that is discussed in the following section.

Validation of the results obtained by new representation
Considering = as well as = + in (6), the λ-generalized gamma function can be expressed as a Fourier transform given below and considering = 1, the generalized gamma function can be expressed as Fourier transform of an arbitrary function ( ), satisfy the following ℱ�√2πℱ[u(t); θ]; ξ� = 2πu(−ξ).
Hence, by applying this on identities (51-52), will lead to the following ℱ�Γ (ν + iθ; a); ξ� = ℱ�√2πℱ�e νx exp�−ae x − be − x ��; ξ� equivalently, which is also obtainable as a specific case of our main result (49) by substituting = ; = + . Furthermore, a substitution = 0 in (55), leads to the following which is also attainable as a precise case of our main result (49). Hence it is testified that the new representation of λ-generalized gamma function produces novel identities, which are unattainable by known techniques but specific forms of new identities are trustworthy with the known methods. Some interesting special cases are for = 1 = Remark 3. It is noticeable that the new obtained integrals contribute only the sum over residues due to the existing poles or singular points in the integrand, which is consistent with the basic result of complex analysis. Next, an application of Parseval's identity of Fourier transform in (54), leads to the following new results about λ-generalized gamma functions Γ (s; a) A substitution = 1 in (59) leads to the following and = 0 leads to the following known result [16,17]

Further properties of the λ-generalized gamma function as a distribution
Here, by taking motivation from [38, Chapter 7], a list of basic properties of the λ-generalized gamma functions are stated and proved. Theorem 3 λ-generalized gamma function holds the subsequent properties as a distribution    where c 1 , and c 2 are arbitrary real or complex constants.
Proof. It can be checked that the methodology to prove (i-vi) is trivial that can be achieved by using the properties of delta function. Therefore, we start proving (vii)

Further Discussion of the class of validity of new representation
Being a singular generalized function, delta function is a linear mapping that maps every function to its value at zero. Due to this property, this new representation has the power to calculate the integrals, which are divergent in the classical sense.
Let us consider (28) and restrict the variable = , to real numbers then we have that can be defined over , that means it is a distribution in ′ because it is convergent for rapidly decreasing and infinitely differentiable functions at 0, such that Next, we take a wider space of infinitely differentiable functions whose derivatives of all order at 0 exist and release the condition of rapidly decreasing. Here we consider some examples Example 1. Let ℘(t) = e ct then ℘ (p) (0) = ; = 0,1,2,3 … 〈Γ (t; a), e ct 〉 = 2π � (− ) (− ) (n − r) p n! r! p! ∞ n,r,p=0 Similarly, ℘(t) = then 〈Γ (t; a), These examples show that new representation of the λ-generalized gamma function is meaningful for all those functions who have derivatives of all orders at 0. This statement can also be generalized as "The new representation of the λ-generalized gamma functions is valid for complex analytic functions at = 0". It is also convergent for all complex analytic functions (who have derivatives of all orders at 0) that also means that example 1-5 are consistent if we consider complex instead of real . Similar results hold for the special cases of the λ-generalized gamma functions i.e, extended gamma, and gamma functions given by Eqs (28), (32) and (36).
As already stated as a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero. Due to this property, new representation has the power to calculate the integrals, which cannot be calculated by using classical method. For example, let ℘(t) = then, It is to be remarked that new representation is convergent for rapidly increasing functions. The integral of rapidly increasing functions is always a challenge nevertheless; this generalized extension of the function has the capacity to do so and it can be defined over the space of rapidly increasing functions. The integral of gamma function is finite so multiplying it with rapidly decreasing function is always convergent. That is trivial to prove. Next, we discuss some further special cases by considering [38, p. 55, problem 10] Therefore, It is meaningful for a class of functions that have derivatives of all orders at point = 0. By using these new representations obtained for the family of gamma functions, it can be observed that all the results that hold for the Laplace transform of delta function, similarly hold for the family of gamma functions, for example Therefore, This gives That yields further, It can be remarked that all the results that hold for delta function can be applied to the family of gamma functions by using this new representation. It is due to the reason that the sum over the coefficients of the new representation is finite and well defined as given in (51). By considering the classical theory of the family of gamma function, for example Eqs (2)-(6), we can note that gamma function has poles at = − but λ-generalized gamma function extends the definition because the exponential factor in the integrand involves parameter > 0. Same fact holds for our new representation, that can be easily proved by taking

Summary and Forthcoming Directions
The combination of distribution theory with different integral transforms is well explored for the analysis of partial differential equations (PDE). Numerous practical questions are impossible to be answered by applying the known techniques but became possible by using this combination. In this paper, a new form of the λ-generalized gamma function is discussed by using delta function so that a new definition of these functions is established for a particular set of test functions. Extensive results are obtained by exploring the details of distributional concepts for λ-generalized gamma function and enlightening their applications for the solution of new problems. As an illustration, we consider the famous Riemann zeta function for the interval 0 < ℜ( ) < 1, as follows λ-generalized gamma function precisely specifies the original gamma function and therefore led to novel outcomes involving different special cases of gamma function. The λ-generalized gamma function and its different special cases are fundamental in different disciplines such as engineering, astronomy and related sciences. Method of computing the new identities involves the desired simplicity. Here we presented only a small number of examples. Further, it is expected that the results obtained in this study will prove significant for further development of λ-generalized gamma function in future work.
The author extends appreciation to the Deanship of Scientific Research at Majmaah University for funding this work under Project Number (RGP-2019-28). The author is also very thankful to the editors and reviewers for their valuable suggestions to improve the manuscript in its present form.