Properties of k-beta function with several variables

Abstract In this paper, we discuss some properties of beta function of several variables which are the extension of beta function of two variables. We define k-beta function of several variables and derive some properties of this function which are the extension of k-beta function of two variables, recently defined by Diaz and Pariguan [4]. Also, we extend the formula Γk(2z) proved by Kokologiannaki [5] via properties of k-beta function.


Introduction
The beta function or Eulerian integral of the first kind with two variables is defined by .x; y/ D 1 Z 0 t x 1 .1 t / y 1 dt ; Re.x/ > 0 ; Re.y/ > 0: (1) The Euler gamma function or Euler integral of the second kind is given by The beta function in terms of gamma function is defined in [8] aš .x; y/ D .x/.y/ .x C y/ ; Re.x/ > 0; Re.y/ > 0: Also, the authors [2] proved some properties of beta function by a simple change of variables aš .x; y/ D 1 Z 0 t x 1 .t C 1/ xCy dt: The collection of most important properties of the beta function of two and more variables is given in [1][2][3]. The definition of beta function is extended to three or more variables in [3]. Let p D .p 1 ; p 2 ; :::; p n / 2 C n ; p i ¤ f0: 1; 2; :::g; n 2. The beta function of n variables is defined aš .p/ Dˇ.p 1 ; p 2 ; :::; p n / D .p 1 /.p 2 /:::.p n / .p 1 C p 2 C ::: C p n / : (6) For n D 2, the classical beta function is represented by integral (1) over a unit interval 0 Ä t Ä 1. If n D 3, we integrate over a triangular region in .x; y/ plane with vertices (0, 0), (1, 0) and (0, 1). This is the set f.x; y/; x 0; y 0; x C y Ä 1g and is called the standard simplex in R 2 . The points .x; y/ of the simplex are (1-1) in correspondence with the triples .x; y; 1 x y/ of non-negative weights with unit sum.The standard simplex in R 3 is f.x; y; z/; x 0; y 0; z 0; x C y C z Ä 1g which is a solid tetrahedron with vertices .0; 0; 0/; .1; 0; 0/.0; 1; 0/ and .0; 0; 1/. In general, we denote the standard simplex in R n by E D E n D f.x 1 ; x 2 ; :::; x n / W x i 0; X x i Ä 1; i D 1; 2; :::; n/g: The points .x 1 ; x 2 ; :::; x n / of the simplex are (1-1) in correspondence with the .n C 1/-tuples .x 1 ; x 2 ; :::; x n ; 1 x 1 x 2 ::: x n / of non-negative weights with unit sum. If n is not mentioned , we write E in place of E n . The interior of E is denoted by i nt .E/ D f.x 1 ; x 2 ; :::; x n / W x i > 0; X x i < 1; i D 1; 2; :::; /g: Let p 2 C n ; p i ¤ 0; jargp i j < 2 and n 2.If E D E n 1 be the standard simplex in R n 1 , then integral form of beta function of n variables defined by the relation (9) and is proved in [3]. .1 x 1 ::: x n 1 / p n 1 dx 1 :::dx n 1 : 2 Properties of beta function of several variables Proposition 2.1. If Re.x/ > 0; Re.x C y/ > 0; Re.x C y C z/ > 0 and Re.w/ > 0, then beta function of four variables can be written asˇ.
Remarks. The relation (10) can be extended up to n variables which will be the representation for beta function of several variables. If there are only two variables, we get the classical definition of beta function. Proof. Using p 1 D p 2 D ::: D p n D 1 n in the relation (6), we havě Re.x C y/ > 0; Re.x C y C z/ > 0 and Re.w/ > 0, then we have the following properties of beta function with three and four variables. andˇ.
x; y; z; w/ D 2 3 Proof. From equation (9), we see that the integral form of beta function for three variables iš .x; y; z/ D Z E p x 1 q y 1 .1 p q/ z 1 dpdq; taking v D 1 p, the above equation takes the form .x; y; z/ D Z p x 1 dp v Z 0 q y 1 .v q/ z 1 dq: Setting q D vt implies dq D vdt and limits of integration becomes 0 to 1. Thus we havě .x; y; z/ D Z p x 1 dp Using the similar reasons, we conclude thať .x; y; z/ Dˇ.y; z/ 1 Z 0 p x 1 .1 p/ yCz 1 dp Dˇ.y; z/ˇ.x; y C z/: Applying the relation (4) on right hand side of above equation, we geť Similarly, we can prove the relation (16).
Remarks. We can extend the results of the Theorem 2.4 up to n variables and that can be expressed aš where .˛/ n D˛.˛C 1/:::.˛C n 1/, is the Pochhammer's symbol.
. Corollary 2.6. If 1 is added to any one of the variables, then following results hold ( only three variables are provided here). pˇ.p; q; r C 1/ D rˇ.p C 1; q; r/ qˇ.p C 1; q; r/ D pˇ.p; q C 1; r/ and qˇ.p; q; r C 1/ D rˇ.p; q C 1; r/: Proof. Just use the definition (6) along with the result .n C 1/ D n.n/.

Main results properties of k-beta function
Recently, Diaz and Pariguan [4] introduced the generalized k-gamma function as and also gave the properties of said function. The k is one parameter of deformation of the classical gamma function such that k ! as k ! 1. The k is based on the repeated appearance of the expression of the following form .˛C k/.˛C 2k/.˛C 3k/:::.˛C .n 1/k/: The function of the variable˛given by the statement (22), denoted by .˛/ n;k , is called the Pochhammer k-symbol.
We obtain the usual Pochhammer symbol .˛/ n by taking k D 1. The deffinition given in (21), is the generalization of .x/ and the integral form of k is given by From (23), we can easily show that The same authors defined the k-beta function aš From the definition ofˇk.x; y/ given in (25) and (26), we can easily prove thať k .x; y/ D 1 kˇ.
Now, we define k-beta function of several variables and prove some properties of the said function. With the help of this definition, we extend some previous results which will help us to compute the values ofˇk function at particular points.
Lemma 3.2. Let and be the positive -finite or a complex measure on a measure space X and T respectively with j j and j j be the corresponding total variation. Assume that the function f W X T ! C is measurable on the product spaceX T with respect to the product -algebra. If the integral Z X Z T jf .x; t /jd j j.x/d j j.t / is finite when evaluated as an iterated integral in one order or the other, then the value of is independent whether it is evaluated as a double integral or an iterated integral in either order.
Remarks. The above Lemma is the Fubini's Theorem. For more complete and precise detail of Fubini's Theorem for positive measure (see Rudin 1974 [13]).
Proof. We denote the integral by I n;k Dˇk.p 1 ; :::; p n /. Thus I 2;k Dˇk.p 1 ; p 2 / which is equivalent to (26). For n 3, first we take the case when p 2 R n ,where R is the set of positive real numbers. The integrand is positive and the integral may be evaluated as an iterated integral by Lemma 3.2. Integrating over x n 1 and defining v D 1 x 1 x 2 ::: x n 2 , we see that Now we make the inductive assumption that I n 1;k is the k-beta function of n 1 variables. So, by the relations (41) and (37), we get I n;k D k n 2ˇk .p 1 ; :::p n 2 ; p n 1 C p n /:kˇk.p n 1 ; p n / D k n 1ˇk .p 1 ; :::; p n /; which is the relation (38) for k > 0; n 3 and p 2 R n . The same proof holds for the case when p 2 C n n f0g if the integrand of the equation (39) can be replaced by the absolute values, simply each p i by Re.p i /; i D 1; 2; :::; n. The iterated integral of the absolute value is finite by the preceding proof for the real case and Lemma 3.2, ensures that the relation (39) is valid for complex case. Now, we prove the symmetry of all arguments. According to the relation (37),ˇk.p 1 ; :::p n / is symmetric in p 1 ; :::; p n and it is worth-while to verify directly that the integral on R.H.S of the equation (38) has the same property. Symmetry in p 1 ; :::; p n 1 is made evident by simply relabeling the variables of integration, so it suffices to prove symmetry in p 1 and p n . We define x n D 1 x 1 ::: and change the variables from x 1 ; :::; x n 1 to x 2 ; :::; x n . The jacobian determinant of the transformation has unit magnitude and the region of integration changes from f.x 1 ; :::; x n 1 / W x 1 0; :::; x n 1 0; x 1 C ::: C x n 1 Ä 1g (43) to f.x 2 ; :::; x n / W x 2 0; :::; x n 0; x 2 C ::: C x n Ä 1g: The last two inequalities in (44) Relabeling x n as x 1 changes it to the original form with p 1 and p n interchanged. Hence the original integral is symmetric in p 1 and p n . We may think of the variables p 1 ; :::; p n as being attached to the vertices of the simplex E D E n 1 as follows: p 1 k to the vertex .1; 0; :::; 0/, p 2 k to .0; 1; :::; 0/,..., p n 1 k to .0; 0; :::; 1/ and p n k to the .0; 0; :::; 0/. Also, the vertex at the origin is at an equal footing with all other vertices. The points .x 1 ; :::; x n 1 / of E n 1 are in (1-1) correspondence with n-tuples .x 1 ; :::; x n 1 ; 1 x 1 ::: x n 1 / of non negative weights with unit sum and the last of these weights is the quantity x n defined in (42) which has the value unity at the vertex .0; 0; :::; 0/. The complete symmetry in 1; :::; n can be exhibited by writing the n 1-fold integral in the relation (38) as n-fold integral with a Dirac delta function in the integranď k .p/ D Now we discuss some properties of k-beta function of several variables.
which is a conclusion of the relation (31).
Remarks. From the corollary 3.5 it seems that k is any natural number. However, we can compute the values of k-beta function for all real numbers k > 0. From the above results, if k D 1, we conclude an important result [2].
Remarks. The relation (49) can be extended up to n variables. Here, we introduced the new notation for beta function of several variables in right hand side of relation (49). If there are only two variables, we get the definition of k-beta function defined in [4] and if k D 1, the classical one.  Similarly, we can prove the relation (54).
Remarks. We can extend the results of Theorem 3.10 up to n variables and can be expressed aš k .x 1 ; x 2 ; :::; x n / D which can be proved like the Theorem 3.10.
Proposition 3.11. If m 1 ; m 2 ; :::; m n 2 N; k > 0 and˛¤ 0, then we have the following properties of the k-beta function with n variables.