A Note on the Beta Function And Some Properties of Its Partial Derivatives

In this paper, the partial derivatives Bp,q(x, y) = ∂ ∂xp∂yq B(x, y) of the Beta function B(x, y) are expressed in terms of a finite number of the Polygamma function, where p and q are non-negative integers, x and y are complex numbers. In particular, Bp,q(x, y) can be expressed by the Riemann zeta function if x is equal to n or n + 1 2 and y is equal to m or m + 1 2 , where n and m are integers. Furthermore, many integral functions associated with B(x, y) and Bp,q(x, y) can be expressed as the closed forms.