Existence of the neutrix limit of the -analogue of the incomplete gamma function and its derivatives

The q -analogue of the incomplete gamma function γ q ( α , x ) is defined for α > 0 , x > 0 and 0 < q < 1 . In this work, the neutrix and the neutrix limit are used to generalize the definition of the q -analogue of the incomplete gamma function and its derivatives for zero and negative integer values of α . We prove the existence of the neutrix limit of the incomplete q -gamma function and its derivatives for all values of α . Recurrence formula for the incomplete q -gamma function at negative integer values is investigated.