The q-analogue of the gamma function is defined by Γ q (x) for x > 0, 0 < q < 1. In this work the neutrix and neutrix limit are used to obtain some equalities of the q-gamma function for all real values of x.

The incomplete beta function B x (a, b) is defined for a, b > 0 and 0 < x < 1. Its definition can be extended, by regularization, to negative non-integer values of a and b. In this paper we define the incomplete beta function B x (a, b) for negative integer values of a and b. Further we prove that the function ∂ m+n ∂a m ∂b n B x (a, b) exists for m, n = 0, 1, 2, . . . and all a and b.

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The distributions (x r + ) −s and (x r + ) −s − were defined as the neutrix limit of the sequences ((x r + ) −s ) n and ((x r + ) −s − ) n respectively for r, s = 1, 2, . . . , see [J.D. Nicholos, B. Fisher, The distribution composition (x r + ) −s , J. Math. Anal. Appl. 258 (2001) 131-145; B. Fisher, On defining the distribution (x r + ) −s − , Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 15 (1985) 119-129]. We here consider these distributions when r = 0. In other words, we define the sth powers of the Heaviside function H (x) in the distributional sense for negative integers. Further compositions are also considered.

The q-beta function B q (t, s) is defined for s, t > 0 and 0 < q < 1. Its definition can be extended, by regularization, to negative non-integer values of t and s. In this paper we define the q-beta function B q (t, s) for negative integer values of t and s.