d-Analog -- from Wolfram MathWorld

d-Analog

The -analog of a complex number is defined as

(1)

(Flajolet et al. 1995). For integer , and

			(2)
			(3)

It can then be extended to complex values via

(4)

(Flajolet et al. 1995). It satisfies the basic functional identity

(5)

The -analog of the polygamma function is

			(6)
			(7)

The first few values are

			(8)
			(9)

			(10)
			(11)

(Flajolet et al. 1995). The first few values are

The -analog of the harmonic numbers is and

			(16)
			(17)

(Flajolet et al. 1995).

The -analog of infinity factorial is given by

(18)

This infinite product can be evaluated in closed form in terms of , the hyperbolic sine , and gamma functions involving roots of unity ,

These are all special cases of a general result for infinite products.

d-Analog