Digital Library of Mathematical Functions
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NIST
10 Bessel FunctionsSpherical Bessel Functions

§10.56 Generating Functions

When 2|t|<|z|,

10.56.1 cosz2-2ztz =coszz+n=1tnn!jn-1(z),
10.56.2 sinz2-2ztz =sinzz+n=1tnn!yn-1(z).
10.56.3 coshz2+2ztz =coshzz+n=1(t)nn!in-1(1)(z),
10.56.4 sinhz2+2ztz =sinhzz+n=1(t)nn!in-1(2)(z),
10.56.5 exp(-z2+2zt)z=-zz+2πn=1(-t)nn!kn-1(z).