About the Project
13 Confluent Hypergeometric FunctionsWhittaker Functions

§13.15 Recurrence Relations and Derivatives

Contents
  1. §13.15(i) Recurrence Relations
  2. §13.15(ii) Differentiation Formulas

§13.15(i) Recurrence Relations

13.15.1 (κμ12)Mκ1,μ(z)+(z2κ)Mκ,μ(z)+(κ+μ+12)Mκ+1,μ(z) =0,
13.15.2 2μ(1+2μ)zMκ12,μ12(z)(z+2μ)(1+2μ)Mκ,μ(z)+(κ+μ+12)zMκ+12,μ+12(z) =0,
13.15.3 (κμ12)Mκ12,μ+12(z)+(1+2μ)zMκ,μ(z)(κ+μ+12)Mκ+12,μ+12(z) =0,
13.15.4 2μMκ12,μ12(z)2μMκ+12,μ12(z)zMκ,μ(z) =0,
13.15.5 2μ(1+2μ)Mκ,μ(z)2μ(1+2μ)zMκ12,μ12(z)(κμ12)zMκ12,μ+12(z) =0,
13.15.6 2μ(1+2μ)zMκ+12,μ12(z)+(z2μ)(1+2μ)Mκ,μ(z)+(κμ12)zMκ12,μ+12(z) =0,
13.15.7 2μ(1+2μ)zMκ+12,μ12(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)+(κμ12)Wκ12,μ+12(z) =0,
13.15.9 Wκ+12,μ12(z)zWκ,μ(z)+(κ+μ12)Wκ12,μ12(z) =0,
13.15.10 2μWκ,μ(z)zWκ+12,μ+12(z)+zWκ+12,μ12(z) =0,
13.15.11 Wκ+1,μ(z)+(2κz)Wκ,μ(z)+(κμ12)(κ+μ12)Wκ1,μ(z) =0,
13.15.12 (κμ12)zWκ12,μ+12(z)+2μWκ,μ(z)(κ+μ12)zWκ12,μ12(z) =0,
13.15.13 (κ+μ12)zWκ12,μ12(z)(z+2μ)Wκ,μ(z)+zWκ+12,μ+12(z) =0,
13.15.14 (κμ12)zWκ12,μ+12(z)(z2μ)Wκ,μ(z)+zWκ+12,μ12(z) =0.

§13.15(ii) Differentiation Formulas

13.15.15 dndzn(e12zzμ12Mκ,μ(z)) =(1)n(2μ)ne12zzμ12(n+1)Mκ12n,μ12n(z),
13.15.16 dndzn(e12zzμ12Mκ,μ(z)) =(12+μκ)n(1+2μ)ne12zzμ12(n+1)Mκ12n,μ+12n(z),
13.15.17 (zddzz)n(e12zzκ1Mκ,μ(z)) =(12+μκ)ne12zznκ1Mκn,μ(z),
13.15.18 dndzn(e12zzμ12Mκ,μ(z)) =(1)n(2μ)ne12zzμ12(n+1)Mκ+12n,μ12n(z),
13.15.19 dndzn(e12zzμ12Mκ,μ(z)) =(1)n(12+μ+κ)n(1+2μ)ne12zzμ12(n+1)Mκ+12n,μ+12n(z),
13.15.20 (zddzz)n(e12zzκ1Mκ,μ(z)) =(12+μ+κ)ne12zzκ+n1Mκ+n,μ(z).
13.15.21 dndzn(e12zzμ12Wκ,μ(z)) =(1)n(12+μκ)ne12zzμ12(n+1)Wκ12n,μ+12n(z),
13.15.22 dndzn(e12zzμ12Wκ,μ(z)) =(1)n(12μκ)ne12zzμ12(n+1)Wκ12n,μ12n(z),
13.15.23 (zddzz)n(e12zzκ1Wκ,μ(z)) =(12+μκ)n(12μκ)ne12zznκ1Wκn,μ(z),
13.15.24 dndzn(e12zzμ12Wκ,μ(z)) =(1)ne12zzμ12(n+1)Wκ+12n,μ+12n(z),
13.15.25 dndzn(e12zzμ12Wκ,μ(z)) =(1)ne12zzμ12(n+1)Wκ+12n,μ12n(z),
13.15.26 (zddzz)n(e12zzκ1Wκ,μ(z)) =(1)ne12zzκ+n1Wκ+n,μ(z).

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