Neumann Polynomial
Polynomials O_n(x) that can be defined by the sum
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
and the generating function
(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) = [画像:(x^2+4)/(x^3)]
(7)
O_3(x) = [画像:(3x^2+24)/(x^4)]
(8)
O_4(x) = [画像:(x^4+16x^2+192)/(x^5)]
(9)
(OEIS A057869).
See also
Neumann Differential Equation, Schläfli PolynomialExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Neumann PolynomialCite this as:
Weisstein, Eric W. "Neumann Polynomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NeumannPolynomial.html