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).

REFERENCES:

Finch, S. R. "Du Bois Reymond's Constants." §3.12 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 237-240, 2003.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 23, 1983.

Sloane, N. J. A. Sequences A085466 and A085467 in "The On-Line Encyclopedia of Integer Sequences."

Watson, G. N. "Du Bois Reymond's Constants." Quart. J. ath. 4, 140-146, 1933.

Young, R. M. "A Rayleigh Popular Problem." Amer. Math. Monthly 93, 660-664, 1986.