Lommel polynomial - Alchetron, The Free Social Encyclopedia

A Lommel polynomial R_m,ν(z), introduced by Eugen von Lommel (1871), is a polynomial in 1/z giving the recurrence relation

J m + ν ( z ) = J ν ( z ) R m , ν ( z ) − J ν − 1 ( z ) R m − 1 , ν + 1 ( z )

where J_ν(z) is a Bessel function of the first kind.

They are given explicitly by

R m , ν = ∑ n = 0 [ m / 2 ] ( − 1 ) n ( m − n ) ! Γ ( ν + m − n ) n ! ( m − 2 n ) ! Γ ( ν + n ) ( z / 2 ) 2 n − m .

References

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