Neumann Polynomial
المؤلف:
Erdelyi, A.; Magnus, W.; Oberhettinger, F.; and Tricomi, F. G.
المصدر:
Higher Transcendental Functions, Vol. 2. Krieger
الجزء والصفحة:
...
21-9-2019
2062
Neumann Polynomial
Polynomials 
that can be defined by the sum
 |
(1)
|
for 
, where 
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]](http://mathworld.wolfram.com/images/equations/NeumannPolynomial/NumberedEquation2.gif) |
(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
(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.
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