Differential Equation of K-Bessel ’ s Function and its Properties

In this paper we solve a differential equation for K Bessel function. We establish a relationship between Bessel function and K -Bessel function. Finally we evaluate the generating function for KBessel function. Mathematics Subject Classification: 33B15, 33C20, 34B30


Introduction
In [1] the authors introduce the generalized K-Gamma Function Γ k (x) as where (x) n,k is the K-Pochhammer symbol and is given by The integral form of the generalized K-Gamma function is given by, from which it follows easily that The Bessel Function (cf.[2], page 109), given as If ϑ is a negative integer, then

Main result
In this section we introduce a differential equation known as K-Bessel differential equation and evaluate its solution which we call as the K-Bessel function.
In theorem 2, we evaluate the functional relation between K -Bessel function and Bessel function.Finally in theorem 4, we find the generating function for K-Bessel function.
Theorem 1.The K-Bessel Differential equation is defined as, Its solution is known as, K-Bessel function and it is written as, where k ∈ R + , ϑ ∈ I and ϑ > −k.
Proof.We can express the solution of (7), in the form of a series of ascending powers of z.Let us assume that its series solution is from which we get and Substituting Eqs. ( 9)-(11) into Eq.( 7), we get Since relation ( 12) is an identity, the coeffieients of various powers of z must be zero.Equating to zero the coefficient of lowest powers of z, i.e. z m−2 in (12), we have Now equating to zero the coefficient of z m−1 in (12), we have Again equating to zero the coefficient of general term, i.e. z m+r in (12), we have Putting r = 1, 3, 5, ....etc in (14), we have a 1 = a 3 = ... = 0(each).
then, we have And for m = − ϑ k , similarly we have Hence.

Theorem 2. The functional relation between K-Bessel function and Bessel function is given by
or the counterpart is and another form or the counterpart is Proof.From defination of K -Bessel function (8), consider the positive sign, using (4) and rearranging the terms we have Hence.
Theorem 3. If ϑ is a negative integer such that ϑ = mk, where m is an integer, then Proof.From ( 16) we have using (6), we obtain using (16), we have Hence.
Theorem 4.(Generating Function for K -Bessel Function) For x = 0 and for all finite z, e Proof.Consider the right side of (21), let us collect powers of z in the summation using relation (20), we have x ϑ J k ϑk (z), using (8), we have For elemintary series manipulation (cf.[2], equation 12, page 58), we have 4) and rearranging the terms Hence.