Neumann polynomial

In mathematics, the Neumann polynomials, introduced by Carl Neumann for the special case $α = 0$ , are a sequence of polynomials in $1 / t$ used to expand functions in term of Bessel functions.

The first few polynomials are

O_{0}^{(α)} (t) = \frac{1}{t},

O_{1}^{(α)} (t) = 2 \frac{α + 1}{t^{2}},

O_{2}^{(α)} (t) = \frac{2 + α}{t} + 4 \frac{(2 + α) (1 + α)}{t^{3}},

O_{3}^{(α)} (t) = 2 \frac{(1 + α) (3 + α)}{t^{2}} + 8 \frac{(1 + α) (2 + α) (3 + α)}{t^{4}},

O_{4}^{(α)} (t) = \frac{(1 + α) (4 + α)}{2 t} + 4 \frac{(1 + α) (2 + α) (4 + α)}{t^{3}} + 16 \frac{(1 + α) (2 + α) (3 + α) (4 + α)}{t^{5}} .

A general form for the polynomial is

O_{n}^{(α)} (t) = \frac{α + n}{2 α} \sum_{k = 0}^{⌊ n / 2 ⌋} (- 1)^{n - k} \frac{(n - k)!}{k!} (\binom{- α}{n - k}) {(\frac{2}{t})}^{n + 1 - 2 k},

and they have the "generating function"

\frac{{(\frac{z}{2})}^{α}}{Γ (α + 1)} \frac{1}{t - z} = \sum_{n = 0} O_{n}^{(α)} (t) J_{α + n} (z),

where J are Bessel functions.

To expand a function f in the form

f (z) = {(\frac{2}{z})}^{α} \sum_{n = 0} a_{n} J_{α + n} (z)

for $| t | < c$ , compute

a_{n} = \frac{Γ (α + 1)}{2 π i} \oint_{| t | = c^{'}} f (t) O_{n}^{(α)} (t) d t,

where $c^{'} < c$ and c is the distance of the nearest singularity of f(z) from $z = 0$ .

Examples

An example is the extension

{(\frac{1}{2} z)}^{s} = Γ (s) \cdot \sum_{k = 0} (- 1)^{k} J_{s + 2 k} (z) (s + 2 k) (\binom{- s}{k}),

or the more general Sonine formula

e^{i γ z} = Γ (s) \cdot \sum_{k = 0} i^{k} C_{k}^{(s)} (γ) (s + k) \frac{J_{s + k} (z)}{{(\frac{z}{2})}^{s}} .

where $C_{k}^{(s)}$ is Gegenbauer's polynomial. Then,^{[citation needed]}^{[original research?]}

\frac{{(\frac{z}{2})}^{2 k}}{(2 k - 1)!} J_{s} (z) = \sum_{i = k} (- 1)^{i - k} (\binom{i + k - 1}{2 k - 1}) (\binom{i + k + s - 1}{2 k - 1}) (s + 2 i) J_{s + 2 i} (z),

\sum_{n = 0} t^{n} J_{s + n} (z) = \frac{e^{\frac{t z}{2}}}{t^{s}} \sum_{j = 0} \frac{{(- \frac{z}{2 t})}^{j}}{j!} \frac{γ (j + s, \frac{t z}{2})}{Γ (j + s)} = \int_{0}^{\infty} e^{- \frac{z x^{2}}{2 t}} \frac{z x}{t} \frac{J_{s} (z \sqrt{1 - x^{2}})}{{\sqrt{1 - x^{2}}}^{s}} d x,

the confluent hypergeometric function

M (a, s, z) = Γ (s) \sum_{k = 0}^{\infty} {(- \frac{1}{t})}^{k} L_{k}^{(- a - k)} (t) \frac{J_{s + k - 1} (2 \sqrt{t z})}{(\sqrt{t z})^{s - k - 1}},

and in particular

\frac{J_{s} (2 z)}{z^{s}} = \frac{4^{s} Γ (s + \frac{1}{2})}{\sqrt{π}} e^{2 i z} \sum_{k = 0} L_{k}^{(- s - 1 / 2 - k)} (\frac{i t}{4}) (4 i z)^{k} \frac{J_{2 s + k} (2 \sqrt{t z})}{{\sqrt{t z}}^{2 s + k}},

the index shift formula

Γ (ν - μ) J_{ν} (z) = Γ (μ + 1) \sum_{n = 0} \frac{Γ (ν - μ + n)}{n! Γ (ν + n + 1)} {(\frac{z}{2})}^{ν - μ + n} J_{μ + n} (z),

the Taylor expansion (addition formula)

\frac{J_{s} (\sqrt{z^{2} - 2 u z})}{{(\sqrt{z^{2} - 2 u z})}^{\pm s}} = \sum_{k = 0} \frac{(\pm u)^{k}}{k!} \frac{J_{s \pm k} (z)}{z^{\pm s}},

(cf.^{[failed verification]}) and the expansion of the integral of the Bessel function,

\int J_{s} (z) d z = 2 \sum_{k = 0} J_{s + 2 k + 1} (z),

are of the same type.

Notes

↑ Abramowitz and Stegun, p. 363, 9.1.82 ff.
↑ Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher Transcendental Functions. Vols. I, II, III, McGraw-Hill, MR 0058756 II.7.10.1, p.64
↑ Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "8.515.1.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 944. ISBN 0-12-384933-0. LCCN 2014010276.

[1] Abramowitz and Stegun, p. 363, 9.1.82 ff.

[2] Erdélyi, Arthur; Magnus, Wilhelm; Oberhettinger, Fritz; Tricomi, Francesco G. (1955), Higher Transcendental Functions. Vols. I, II, III, McGraw-Hill, MR 0058756 II.7.10.1, p.64

[Zwillinger_2014-3] Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. "8.515.1.". In Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. p. 944. ISBN 0-12-384933-0. LCCN 2014010276.

Neumann polynomial

Examples

See also

Notes