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13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.15 Recurrence Relations and Derivatives

Contents

§13.15(i) Recurrence Relations

13.15.1 (κ-μ-12)Mκ-1,μ(z)+(z-2κ)Mκ,μ(z)+(κ+μ+12)Mκ+1,μ(z) =0,
13.15.2 2μ(1+2μ)zMκ-12,μ-12(z)-(z+2μ)(1+2μ)Mκ,μ(z)+(κ+μ+12)zMκ+12,μ+12(z) =0,
13.15.3 (κ-μ-12)Mκ-12,μ+12(z)+(1+2μ)zMκ,μ(z)-(κ+μ+12)Mκ+12,μ+12(z) =0,
13.15.4 2μMκ-12,μ-12(z)-2μMκ+12,μ-12(z)-zMκ,μ(z) =0,
13.15.5 2μ(1+2μ)Mκ,μ(z)-2μ(1+2μ)zMκ-12,μ-12(z)-(κ-μ-12)zMκ-12,μ+12(z) =0,
13.15.6 2μ(1+2μ)zMκ+12,μ-12(z)+(z-2μ)(1+2μ)Mκ,μ(z)+(κ-μ-12)zMκ-12,μ+12(z) =0,
13.15.7 2μ(1+2μ)zMκ+12,μ-12(z)-2μ(1+2μ)Mκ,μ(z)+(κ+μ+12)zMκ+12,μ+12(z) =0.
13.15.8 Wκ+12,μ+12(z)-zWκ,μ(z)+(κ-μ-12)Wκ-12,μ+12(z) =0,
13.15.9 Wκ+12,μ-12(z)-zWκ,μ(z)+(κ+μ-12)Wκ-12,μ-12(z) =0,
13.15.10 2μWκ,μ(z)-zWκ+12,μ+12(z)+zWκ+12,μ-12(z) =0,
13.15.11 Wκ+1,μ(z)+(2κ-z)Wκ,μ(z)+(κ-μ-12)(κ+μ-12)Wκ-1,μ(z) =0,
13.15.12 (κ-μ-12)zWκ-12,μ+12(z)+2μWκ,μ(z)-(κ+μ-12)zWκ-12,μ-12(z) =0,
13.15.13 (κ+μ-12)zWκ-12,μ-12(z)-(z+2μ)Wκ,μ(z)+zWκ+12,μ+12(z) =0,
13.15.14 (κ-μ-12)zWκ-12,μ+12(z)-(z-2μ)Wκ,μ(z)+zWκ+12,μ-12(z) =0.

§13.15(ii) Differentiation Formulas

13.15.15 dndzn(e12zzμ-12Mκ,μ(z)) =(-1)n(-2μ)ne12zzμ-12(n+1)Mκ-12n,μ-12n(z),
13.15.16 dndzn(e12zz-μ-12Mκ,μ(z)) =(12+μ-κ)n(1+2μ)ne12zz-μ-12(n+1)Mκ-12n,μ+12n(z),
13.15.17 (zddzz)n(e12zz-κ-1Mκ,μ(z)) =(12+μ-κ)ne12zzn-κ-1Mκ-n,μ(z),
13.15.18 dndzn(e-12zzμ-12Mκ,μ(z)) =(-1)n(-2μ)ne-12zzμ-12(n+1)Mκ+12n,μ-12n(z),
13.15.19 dndzn(e-12zz-μ-12Mκ,μ(z)) =(-1)n(12+μ+κ)n(1+2μ)ne-12zz-μ-12(n+1)Mκ+12n,μ+12n(z),
13.15.20 (zddzz)n(e-12zzκ-1Mκ,μ(z)) =(12+μ+κ)ne-12zzκ+n-1Mκ+n,μ(z).
13.15.21 dndzn(e12zz-μ-12Wκ,μ(z)) =(-1)n(12+μ-κ)ne12zz-μ-12(n+1)Wκ-12n,μ+12n(z),
13.15.22 dndzn(e12zzμ-12Wκ,μ(z)) =(-1)n(12-μ-κ)ne12zzμ-12(n+1)Wκ-12n,μ-12n(z),
13.15.23 (zddzz)n(e12zz-κ-1Wκ,μ(z)) =(12+μ-κ)n(12-μ-κ)ne12zzn-κ-1Wκ-n,μ(z),
13.15.24 dndzn(e-12zz-μ-12Wκ,μ(z)) =(-1)ne-12zz-μ-12(n+1)Wκ+12n,μ+12n(z),
13.15.25 dndzn(e-12zzμ-12Wκ,μ(z)) =(-1)ne-12zzμ-12(n+1)Wκ+12n,μ-12n(z),
13.15.26 (zddzz)n(e-12zzκ-1Wκ,μ(z)) =(-1)ne-12zzκ+n-1Wκ+n,μ(z).

Other versions of several of the identities in this subsection can be constructed by use of (13.3.29).