DLMF:12.13.E6 (Q4235)

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DLMF:12.13.E6
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    Statements

    Defining formula
    n ! ⁢ U ⁡ ( n + 1 2 , z ) = i n ⁢ e - 1 2 ⁢ z 2 ⁢ erfc ⁡ ( z / 2 ) ⁢ U ⁡ ( - n - 1 2 , i ⁢ z ) + ∑ m = 1 ⌊ 1 2 ⁢ n + 1 2 ⌋ U ⁡ ( 2 ⁢ m - n - 1 2 , z ) , 𝑛 parabolic-U 𝑛 1 2 𝑧 superscript 𝑖 𝑛 superscript 𝑒 1 2 superscript 𝑧 2 complementary-error-function 𝑧 2 parabolic-U 𝑛 1 2 𝑖 𝑧 superscript subscript 𝑚 1 1 2 𝑛 1 2 parabolic-U 2 𝑚 𝑛 1 2 𝑧 {\displaystyle{\displaystyle n!U\left(n+\tfrac{1}{2},z\right)=i^{n}e^{-\frac{1% }{2}z^{2}}\operatorname{erfc}(z/\sqrt{2})U\left(-n-\tfrac{1}{2},iz\right)+\sum% _{m=1}^{\left\lfloor\frac{1}{2}n+\frac{1}{2}\right\rfloor}U\left(2m-n-\tfrac{1% }{2},z\right),}}
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    DLMF id
    DLMF:12.13.E6
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    constraint
    n = 0 , 1 , 2 , … . 𝑛 0 1 2 … {\displaystyle{\displaystyle n=0,1,2,\dots.}}
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    Symbols used
    complementary error function
    Defining formula
    erfc ⁡ z complementary-error-function 𝑧 {\displaystyle{\displaystyle\operatorname{erfc}\NVar{z}}}
    xml-id
    C7.S2.E2.m2adec
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    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2aedec
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    Q10755
    Defining formula
    ! {\displaystyle{\displaystyle!}}
    xml-id
    introduction.Sx4.p1.t1.r15.m5abdec
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    Q11169
    Defining formula
    ⌊ x ⌋ 𝑥 {\displaystyle{\displaystyle\left\lfloor\NVar{x}\right\rfloor}}
    xml-id
    introduction.Sx4.p1.t1.r17.m4adec
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    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2adec
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    Q11320
    Defining formula
    U ⁡ ( a , z ) parabolic-U 𝑎 𝑧 {\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}
    xml-id
    C12.S2.SS1.p1.m1acdec
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