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10 Bessel FunctionsKelvin Functions

§10.63 Recurrence Relations and Derivatives

Contents
  1. §10.63(i) berνx, beiνx, kerνx, keiνx
  2. §10.63(ii) Cross-Products

§10.63(i) berνx, beiνx, kerνx, keiνx

Let fν(x), gν(x) denote any one of the ordered pairs:

10.63.1 berνx,beiνx;
beiνx,berνx;
kerνx,keiνx;
keiνx,kerνx.

Then

10.63.2 fν1(x)+fν+1(x) =(ν2/x)(fν(x)gν(x)),
fν+1(x)+gν+1(x)fν1(x)gν1(x)=22fν(x),
fν(x) =(1/2)(fν1(x)+gν1(x))(ν/x)fν(x),
fν(x) =(1/2)(fν+1(x)+gν+1(x))+(ν/x)fν(x).
10.63.3 2berx =ber1x+bei1x,
2beix =ber1x+bei1x.
10.63.4 2kerx =ker1x+kei1x,
2keix =ker1x+kei1x.

§10.63(ii) Cross-Products

Let

10.63.5 pν =berν2x+beiν2x,
qν =berνxbeiνxberνxbeiνx,
rν =berνxberνx+beiνxbeiνx,
sν =(berνx)2+(beiνx)2.

Then

10.63.6 pν+1 =pν1(4ν/x)rν,
qν+1 =(ν/x)pν+rν=qν1+2rν,
rν+1 =((ν+1)/x)pν+1+qν,
sν =12pν+1+12pν1(ν2/x2)pν,

and

10.63.7 pνsν=rν2+qν2.

Equations (10.63.6) and (10.63.7) also hold when the symbols ber and bei in (10.63.5) are replaced throughout by ker and kei, respectively.