Neumann Polynomial

Polynomials  that can be defined by the sum

(1)

for , where  is the floor function. They obey the recurrence relation

(2)

for . They have the integral representation

(3)

and the generating function

(4)

(Gradshteyn and Ryzhik 2000, p. 990), and obey the Neumann differential equation.

The first few Neumann polynomials are given by

			(5)
			(6)
			(7)
			(8)
			(9)

(OEIS A057869).

REFERENCES:

Erdelyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G. Higher Transcendental Functions, Vol. 2. Krieger, pp. 32-33, 1981.

Gradshteyn, I. S. and Ryzhik, I. M. "Neumann's and Schläfli Polynomials:  and ." §8.59 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, pp. 989-991, 2000.

Sloane, N. J. A. Sequence A057869 in "The On-Line Encyclopedia of Integer Sequences."

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 196, 1993.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, pp. 298-305, 1966.