Author : Hamzah Farhood, Jinan
journal of kerbala university,
Volume 8, Issue 4, Pages 120-130
In this paper, we consider the probability density function (pdf) of a non-central distribution with odd number of degrees of freedom n. This pdf is represented in the literature as an infinite sum. Kettani  presented two alternative expressions to this pdf. The first expression is in terms of the partial derivative of the hyperbolic cosine function and the second expression, on the other hand, is a finite sum representation of terms only instead of the infinite sum.
In this paper, we prove Theorem 3.4 that implies if m = 0 which introduced by Kettani .
Also, before the end section of this paper, we present four different theorems includes the pdf for the non-central distribution which are general recurrence relation for such pdf.