Neumann polynomial

An example is the extension

:$\backslash left(\backslash tfrac\{1\}\{2\}z\backslash right)^s=\; \backslash Gamma(s)\backslash cdot\backslash sum\_\{k=0\}(-1)^k\; J\_\{s+2k\}(z)(s+2k)\{-s\; \backslash choose\; k\},$

or the more general Sonine formula{{citation

| last1 = Erdélyi | first1 = Arthur

| last2 = Magnus | first2 = Wilhelm

| last3 = Oberhettinger | first3 = Fritz

| last4 = Tricomi | first4 = Francesco G.

| mr = 58756

| publisher = McGraw-Hill

| title = Higher Transcendental Functions. Vols. I, II, III

| year = 1955}} II.7.10.1, p.64

:$e^\{i\; \backslash gamma\; z\}=\; \backslash Gamma(s)\backslash cdot\backslash sum\_\{k=0\}i^k\; C\_k^\{(s)\}(\backslash gamma)(s+k)\backslash frac\{J\_\{s+k\}(z)\}\{\backslash left(\backslash frac\; z\; 2\backslash right)^s\}.$

where $C\_k^\{(s)\}$ is Gegenbauer's polynomial. Then,{{fact|date=September 2011}}{{or|date=September 2011}}

:$\backslash frac\{\backslash left(\backslash frac\; z\; 2\backslash right)^\{2k\}\}\{(2k-1)!\}J\_s(z)=\; \backslash sum\_\{i=k\}(-1)^\{i-k\}\{i+k-1\backslash choose\; 2k-1\}\{i+k+s-1\backslash choose\; 2k-1\}(s+2i)J\_\{s+2i\}(z),$

:$\backslash sum\_\{n=0\}\; t^n\; J\_\{s+n\}(z)=\; \backslash frac\{e^\{\backslash frac\{t\; z\}2\}\}\{t^s\}\; \backslash sum\_\{j=0\}\backslash frac\{\backslash left(-\backslash frac\{z\}\{2t\}\backslash right)^j\}\{j!\}\backslash frac\{\backslash gamma\; \backslash left(j+s,\backslash frac\{t\; z\}\{2\}\backslash right)\}\{\backslash ,\backslash Gamma\; (j+s)\}=\; \backslash int\_0^\backslash infty\; e^\{-\backslash frac\{z\; x^2\}\{2\; t\}\}\backslash frac\; \{z\; x\}\{t\}\; \backslash frac\{J\_s(z\backslash sqrt\{1-x^2\})\}\{\backslash sqrt\{1-x^2\}^s\}\backslash ,dx,$

the confluent hypergeometric function

:$M(a,s,z)=\; \backslash Gamma\; (s)\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash left(-\backslash frac\{1\}\{t\}\backslash right)^k\; L\_k^\{(-a-k)\}(t)\; \backslash frac\{J\_\{s+k-1\}\backslash left(2\; \backslash sqrt\{t\; z\}\backslash right)\}\{(\backslash sqrt\{t\; z\})^\{s-k-1\}\},$

and in particular

:$\backslash frac\{J\_s(2\; z)\}\{z^s\}=\; \backslash frac\{4^s\; \backslash Gamma\backslash left(s+\backslash frac12\backslash right)\}\{\backslash sqrt\backslash pi\}e^\{2\; i\; z\}\backslash sum\_\{k=0\}L\_k^\{(-s-1/2-k)\}\backslash left(\backslash frac\{it\}4\backslash right)(4\; i\; z)^k\; \backslash frac\{J\_\{2s+k\}\backslash left(2\backslash sqrt\{t\; z\}\backslash right)\}\{\backslash sqrt\{t\; z\}^\{2s+k\}\},$

the index shift formula

:$\backslash Gamma(\backslash nu-\backslash mu)\; J\_\backslash nu(z)=\; \backslash Gamma(\backslash mu+1)\; \backslash sum\_\{n=0\}\backslash frac\{\backslash Gamma(\backslash nu-\backslash mu+n)\}\{n!\backslash Gamma(\backslash nu+n+1)\}\; \backslash left(\backslash frac\; z\; 2\backslash right)^\{\backslash nu-\backslash mu+n\}J\_\{\backslash mu+n\}(z),$

the Taylor expansion (addition formula)

:$\backslash frac\{J\_s\backslash left(\backslash sqrt\{z^2-2uz\}\backslash right)\}\{\backslash left(\backslash sqrt\{z^2-2uz\}\backslash right)^\{\backslash pm\; s\}\}=\; \backslash sum\_\{k=0\}\backslash frac\{(\backslash pm\; u)^k\}\{k!\}\backslash frac\{J\_\{s\backslash pm\; k\}(z)\}\{z^\{\backslash pm\; s\}\},$

(cf.{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=Academic Press, Inc. |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=0-12-384933-0 |lccn=2014010276 |title-link=Gradshteyn and Ryzhik |chapter=8.515.1. |page=944}}{{verification failed|date=September 2011|reason=The referenced identity seems to be only superficially similar. It does not directly support the identity here.}}) and the expansion of the integral of the Bessel function,

:$\backslash int\; J\_s(z)dz=\; 2\; \backslash sum\_\{k=0\}\; J\_\{s+2k+1\}(z),$

are of the same type.

• Bessel function
• Bessel polynomial
• Lommel polynomial
• Hankel transform
• Fourier–Bessel series

Category:Polynomials

Category:Special functions