du Bois-Reymond Constants

The constants defined by

(1)

These constants can also be written as the sums

(2)

and

(3)

(E. Weisstein, Feb. 3, 2015), where is the th positive root of

(4)

and is the sinc function.

diverges, with the first few subsequent constant numerically given by

			(5)
			(6)
			(7)

Rather surprisingly, the even-ordered du Bois Reymond constants (and, in particular, ; Le Lionnais 1983) can be computed analytically as polynomials in ,

			(8)
			(9)
			(10)

(OEIS A085466 and A085467) as found by Watson (1933). For positive integer , these have the explicit formula

(11)

where denotes a complex residue and is a Kronecker delta (V. Adamchik).