G.t'Hooft lecture Special Functions and Polynomials

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1、SPECIALFUNCTIONSandPOLYNOMIALSGerard’tHooftStefanNobbenhuisInstituteforTheoreticalPhysicsUtrechtUniversity,Leuvenlaan43584CCUtrecht,theNetherlandsandSpinozaInstitutePostbox80.1953508TDUtrecht,theNetherlandsManyofthespecialfunctionsandpolynomialsareconstructedalongstandardprocedu

2、resInthisshortsurveywelistthemostessentialones.October4,200511LegendrePolynomialsP`(x).DifferentialEquation:2000(1−x)P`(x)−2xP`(x)+`(`+1)P`(x)=0,ord2d(1−x)P`(x)+`(`+1)P`(x)=0.(1.1)dxdxGeneratingfunction:X∞`2−1P`(x)t=(1−2xt+t)2for

3、t

4、<1,

5、x

6、≤1.(1.2)`=0Orthonormality:Z12P`(x)P`0(x)dx

7、=δ``0,(1.3)-12`+1X∞00P`(x)P`(x)(2`+1)=2δ(x−x).(1.4)`=0ExpressionsforP`(x):1[X`/2](−1)ν(2`−2ν)!`−2νP`(x)=x(1.5)2`ν!(`−ν)!(`−2ν)!ν=01d`2`=(x−1),(1.6)`!2`dx1Zπ√=(x+x2−1cosϕ)`dϕ.(1.7)π0Recurrencerelations:`P`−1−(2`+1)xP`+(`+1)P`+1=0;x2−10P`=xP`−1+P`−1;`00xP`−`P`=P`−1;00xP`+(`+1)P`

8、=P`+1;d[P`+1−P`−1]=(2`+1)P`.(1.8)dxExamples:P=1,P=x,P=1(3x2−1),P=1x(5x2−3).(1.9)01223212AssociatedLegendreFunctionsPm(x).`Differentialequation:m22m00m0m(1−x)P`(x)−2xP`(x)+`(`+1)−2P`(x)=0.(2.1)1−xGeneratingfunction:X∞X`Pm(x)zmy`h√i−1`=1−2yx+z1−x2+y22.(2.2)m!`=0m=0Orthogonality

9、:Z1mm2(`+m)!0P`(x)P`0(x)dx=δ``0,(`,`≥m).(2.3)-12`+1(`−m)!X∞(`−m)!mm000(2`+1)P`(x)P`(x)=2δ(x−x),(

10、x

11、<1and

12、x

13、<1).(2.4)(`+m)!`=mExpressionsforPm(x)1:`!mm21mdP`(x)=(1−x)2P`(x).(2.5)dx(`+m)!Zπ√`Pm(x)=(−1)m/2x+x2−1cosϕcosmϕdϕ.(2.6)``!π0Recurrencerelations:m+1√2mxmm−1P`−2P`+{`(`+1)−m

14、(m−1)}P`=0(2.7)1−x√1−x2Pm+1(x)=(1−x2)Pm(x)0+mxPm(x),```mmm(2`+1)xP`=(`+m)P`−1+(`+1−m)P`+1,(2.8)√xPm=Pm−(`+1−m)1−x2Pm−1,``−1`√Pm−Pm=(2`+1)Pm−11−x2,(2.9)`+1`−1`andvariousothers.Examples:√P1=1−x2,P2=3(1−x2),1√2P1=3x1−x2,P2=15x(1−x2).(2.10)231NotethatsomeauthorsdefinePm(x)withafactor

15、(−1)m,givingPm(x)=(−1)m(1−
m``21mdx)2P`(x).Obviouslythisminussignpropagatestothegeneratingfunction,therecurrencedxrelationsandtheexplicitexamples,whenmisodd.23BesselJn(x)andHankelHn(x)functions.Differentialequation(forbothJnandHn):200022xJn(x)+xJn(x)+(x−n)Jn(x)=0.(3.1)Generating

16、function(ifninteger):X∞snxα2(s−)Jn(αx)=e2s,(3