Two-dimensional generalized gamma function and its applications

In this article, we present a new two-dimensional generalization of the gamma function based on the product of the one-dimensional generalized beta function and the one-dimensional generalized gamma function. As will become clear later, this generalization is also a generalization the famous formula that gives the connection between the classical gamma and beta functions. Next we present properties of this generalization, some series for the generalized beta function. As a practical application of the two-dimensional generalized gamma function, we will show how it can be used to represent a fairly wide class of double integrals in the form of functional series.


Introduction of a two-dimensional generalization of the gamma function and study of its properties.
We will consider a one-dimensional generalization of the gamma function in the form * g x g x x e dx ( ) ( ): ( ) one-dimensional generalization of the beta function in the form and two-dimensional generalization of the Gamma function Q !0 , Z ! 0, J t 0 .Obviously, if : { 1 and J 0 , then our generalized gamma function is transformed into the product of two classical gamma functions: * * ( ) ( ) Q Z .In this article we will set the task of finding the product of generalizations ( 1) and ( 2) in the form (3).That is, we will find for the function Ω such a condition that this product looks like a special case of the right-hand side of the formula (3).Thus, we obtain a generalization of the well-known formula for the relationship between the classical gamma function and the beta function To do this, we will consider the following generalizations of the gamma function of the form and provided that integrals (4), ( 2) exist, and we will obtain a double integral, which will be equal to their product.We will also consider integrals ; (5) ( , ; ) : ( , ; ) :

g x y y x x y e dxdy
x y where D ! 0, E ! 0, O t 0 for all three previous integrals.We will see later in our study that in the generalization (3) it is convenient to choose the order of variables Ω( , ) y x , and not Ω( , ) x y .Let us introduce the following notation Theorem 1.Let us assume that Q x y ( , ) is a given function satisfying the following conditions: 1) the function Q x y ( , ) exists, is non-negative, continuous at all points of the region T x y R x y t t ^, : , 2 0 0 , with the possible exception of a finite number of points M 1 , M 2 , …, M κ in this region.At points M 1 , M 2 , …, M κ the function Q x y ( , ) tends to f ; 2) integrals ( 4), ( 2), ( 5), ( 6), (7) exist and converge for all values D ! 0 , E ! 0 , O t 0 ; 3) the following equality holds ( , ; ) Then the following formula will be valid  it's from 0 to f .Then we obtain ), we obtain . ◊ Next, we will obtain a similar theorem for the case of an alternating function Q .For this we will consider the integrals ( , ; ) : f y x y g x y y x x y e dydx x y x y R x y ( , ; ) : x y e dydx x y and ( , ; ) : x y e dxdy x y where D ! 0 , E ! 0 , O t 0 for all three previous integrals.

Theorem 2. Let us assume that
is a given function satisfying the following conditions: 1) the function Q x y ( , ) exists and is continuous at all points of the region : , 2 0 0 , with the possible exception of a finite number of points ( , ) tends to f or to f ; 2) integrals ( 12), ( 13), ( 14), (15), (16) exist and converge for all values D ! 0 , E ! 0 , O t 0 ; 3) the equality (9) holds.Then the formula (10) will be valid.Remark 1. Theorems 1 and 2 will be valid if in condition 1) we replace the continuity of function Next, we will consider the case of functions that are Riemann integrable.

Theorem 3. Let us assume that
is a given function satisfying the following con- ditions: the functions . Then the formula (10) will be valid.Proof.The necessity of the conditions of the theorem here is obvious.
( , ; ) : Here we should show that double integrals, when changing the order of integration, will give the same result.
Next, changing the order of integration we obtain ( , ; ) Next, we get Substitution ( 22) into (21), we to obtain formula (9).Thus, under the assumptions of this theorem, an integral of the form (6) does not depend on the order of integration.◊ Next, we will consider the case of a continuous function Q .
Next, we will introduce the definition of the generalized two-dimensional gamma function in the form of a double integral.One of the main properties of this function is a generalization of the wellknown formula for the relationship between the classical gamma function and the beta function.

Definition 1. Let's the function
satisfy the conditions of at least one of Theorems 1-4.We define the two-dimensional generalized gamma function as a double integral as follows: ( , ; ) :

ФІЗИКО-МАТЕМАТИЧНІ НАУКИ
Theorem 5 (Properties of the two-dimensional generalized gamma function).We assume that the conditions of at least one of Theorems 1-4 are satisfied.Then for values of D ! 0 , E ! 0 , O t 0 , we obtain the following properties of the two-dimensional generalized gamma function: x y e dxdy x y e dxdy

Generalized formula for the relationship between the gamma function and the beta function
5.1.If we assume f ≡ 1 , g ≡ 1 , O 0 in the formula ( 24), then we obtain formula for the relationship between the classical gamma function and the beta function (25) 5.2.If we assume f ≡ 1 , g ≡ 1 , D 1 , E 1 in the formula (24), then we obtain classical gamma function (26) 5. 3

. If we assume
/ , E 1 2 / in the formula (24), then we also find a special case of transforming the two-dimensional generalized gamma function into the classical gamma function (27) 5.4.If we assume f ≡ 1 , g ≡ 1 in the formula (24), then we obtain y x x y e dydx B x y 6.If the function g identically not equal to zero, then Proof.Properties 1-4 are obvious.Property 5 follows from Definition 1 of the two-dimensional generalized gamma function (23) and the formula (10), due to the fulfillment of the conditions of at least one of Theorems 1-4.Property 7 follows from Property 5 and the following formula, which is satisfied under the same assumptions Lemma 1.Let us assume that following conditions are met: 1) the function ′ f x ( ) exists on ( , )  0 1 and the following integral exists 2) the following limits are valid , if x o 0 and x o 1 , D ! 0, E ! 0. Then the following formula will be valid (30) Proof.We obtain Equating the right sides of formulas (31) end (32), we obtain (30).◊ Lemma 2. Let us assume that following conditions are met: 1.) conditions 1) and 2) of Lemma 1 are satisfied; satisfy the conditions of at least one of Theorems 1-4.
Then the following formulas will be valid ( , ; ) Proof.By multiplying equality (30) by * g ( ) ( ) D E O 1 , and then based on the definition of the two-dimensional generalized gamma function, we find (31).

Lemma 3. Let's the function
satisfy the conditions of at least one of Theorems 1-4.
Then for ∀ D ! 0 , E ! 0 , O t 0 and for ∀ l , m , n , where l m n N , ,

^` ^0 , exist derivatives of all orders for the two-dimensional generalized gamma function
( , ; )

y e dydx x y O
, Theorem 6 (Formula for differentiating the two-dimensional generalized gamma function).Let's the satisfy the conditions of at least one of Theorems 1-4.Then the following formula will be valid .
In the notation style, according to Definition 1, our differentiating formula (35) has the form Proof.   ( , ) .
In this proof we applied the transformations , from formula (36), we find ( , ) , where Remark 4. Assuming f ≡ 1 , g ≡ 1 , O 0 in the formula (36), we obtain a formula for differentiating the classical gamma and beta functions of any order where D ! 0 , E ! 0 , m n N , ^` ^0 .The formula (37) is also a generalization of formula for the relation- ship between the classical gamma function and the beta function , if m n = = 0 .Remark 5. We will assume that 9 U 9 U 9 U ^ R , then there are all these values belonging to the set of real numbers.Theorems 1-6, Lems 1-3 and Remark 1-4 will be valid if we consider our values of α , β , λ in the complex plane, provided that D 9 U i , 9 3 0 t .

Practical applications of the generalized two-dimensional gamma function for calculating double integrals, including those containing some special functions.
In this chapter, we will present several theorems containing integral equalities using the two-dimensional generalized gamma function.
Theorem 7. Let's the function f y x y § © ¨• ¹ ¸ satisfy the conditions of at least one of Theorems 1-4, provided that g ≡ 1 .Then for any a > 0 , D ! 0 , E ! 0 , O t 0 the following formulas will be valid: x y e bx y dydx x y e bx y dydx Proof.We obtain x y e bx y dydx Equating the real and imaginary parts of equalities (40), (41), we obtain (38), (39), with further consideration of the following two equalities at D ! 0 , E ! 0 , O t 0 : , a > 0 satisfy the conditions of function Q x y ( , ) of at least one of Theorems 1-4.Then for any a > 0 , D ! 0 , E ! 0 , O t 0 the following formulas will be valid:  Proof is similar to Theorem 7. ◊ One of the more general possibilities for the practical application of our generalized two-dimensional gamma-function is provided by the following theorem.
Theorem 9. Let us assume that following conditions are met: satisfy the conditions of at least one of Theorems 1-4; 2) The function f x is expanded into a Taylor series , A f .Then for any D ! 1 , E ! 1 , O t 0 the following formulas will be valid: Proof.Multiplying both sides of equality (42) by , and integrating from 0 to 1, we obtain .
We find the last two formulas using the definition of the two-dimensional generalized gamma function and generalized formula (24) for the relationship between the gamma function and the beta function.◊ Remark 6. Obviously, all theorems, lemmas and remarks of this section will be valid if we consider the case ¸ , we can easily extend all the result to this case.
As an additional application, we can consider the hypergeometric function of the form [3]: O t 0 , z is not a real number greater than or equal to 1. Obviously, formulas (46) and (47) for the generalized hypergeometric function (45) give us many options for choosing function g and parameter λ .
Conclusion.In the article we showed that a special case (23) of the two-dimensional generalized gamma function (3) is the product of the one-dimensional generalized beta function (2) and the one-dimensional generalized gamma function (4).Thus, formula (24) for this special case (23) is a generalization of formula for the relationship between the classical gamma function and beta function.We also obtained quite a few properties of this generalization of the two-dimensional gamma function, including a formula for its differentiation of any order.Next, we showed some practical applications of this generalization, including its use for transforming a one-dimensional generalized hypergeometric function into a two-dimensional generalized gamma function.

F,F
conditions of function Q x y ( , ) of at least one of Theorems 1-4, taking into account the Remark 5. Then we obtain Re( ) O t 0 , z is not a real number greater than or equal to 1, the function g identically not equal to zero.If we assume g ≡ 1 , and the function Q x it satisfy the conditions of function Q x y ( , ) of at least one of Theorems 1-4, we obtain