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)

where  is the digamma function.

The -analog of the Euler-Mascheroni constant  is

			(10)
			(11)

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

			(12)
			(13)
			(14)
			(15)

where  is a harmonic number.

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 ,

			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)
			(26)
			(27)

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

REFERENCES:

Flajolet, P.; Labelle, G.; Laforest, L.; and Salvy, B. "Hypergeometrics and the Cost Structure of Quadtrees." Random Structure Alg. 7, 117-144, 1995. http://algo.inria.fr/flajolet/Publications/publist.html.