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Neumann Polynomial

Polynomials O_n(x) that can be defined by the sum

O_n(x)=1/4sum_(k=0)^(|_n/2_|)(n(n-k-1)!)/(k!)(1/2x)^(2k-n-1)
(1)

for n>=1, where |_x_| is the floor function. They obey the recurrence relation

O_n(x)=-n/(n-2)O_(n-2)(x)+(2n)/xO_(n-1)(x)+(2(n-1))/((n-2)x)sin^2[1/2(n-1)pi]
(2)

for n>=3. They have the integral representation

O_n(x)=int_0^infty((u+sqrt(u^2+x^2))^n+(u-sqrt(u^2+x^2))^n)/(2x^(n+1))e^(-u)du,
(3)

and the generating function

1/(x-xi)=J_0(xi)x^(-1)+2sum_(n=1)^inftyJ_n(xi)O_n(x)
(4)

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

The first few Neumann polynomials are given by

O_0(x)=1/x
(5)
O_1(x)=1/(x^2)
(6)
O_2(x)=(x^2+4)/(x^3)
(7)
O_3(x)=(3x^2+24)/(x^4)
(8)
O_4(x)=(x^4+16x^2+192)/(x^5)
(9)

(OEIS A057869).

See also

Neumann Differential Equation, Schläfli Polynomial

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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: O_n(z) and S_n(z)." §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.

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Neumann Polynomial

Cite this as:

Weisstein, Eric W. "Neumann Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NeumannPolynomial.html

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