Digital Library of Mathematical Functions
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10 Bessel FunctionsKelvin Functions

§10.63 Recurrence Relations and Derivatives

Contents

§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 fν-1(x)+fν+1(x) =-(ν2/x)(fν(x)-gν(x)),
fν+1(x)+gν+1(x)-fν-1(x)-gν-1(x) =22fν(x),
fν(x) =-(1/2)(fν-1(x)+gν-1(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 =berνxbeiνx-berνxbeiνx,
rν =berνxberνx+beiνxbeiνx,
sν =(berνx)2+(beiνx)2.

Then

10.63.6 pν+1 =pν-1-(4ν/x)rν,
qν+1 =-(ν/x)pν+rν
=-qν-1+2rν,
rν+1 =-((ν+1)/x)pν+1+qν,
sν =12pν+1+12pν-1-(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.