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1: 12.1 Special Notation

… ►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. … ►Whittaker’s notation $D_{\nu }\left(z\right)$ is useful when $\nu$ is a nonnegative integer (Hermite polynomial case).

2: 18.1 Notation

… ►They are defined in the literature by $C_0^{(0)}\left(x\right)=1$ and …

3: 15.2 Definitions and Analytical Properties

… ►For example, when $a=-m$, $m=0,1,2,\mathrm{\dots }$, and $c\ne 0,-1,-2,\mathrm{\dots }$, $F(a,b;c;z)$ is a polynomial: …

4: 16.2 Definition and Analytic Properties

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Polynomials

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5: 12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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6: 13.2 Definitions and Basic Properties

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13.2.9 $$U(a,n+1,z)=\frac{{(-1)}^{n+1}}{n!\mathrm{\Gamma }\left(a-n\right)}\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_k}{{\left(n+1\right)}_{k}k!}{z}^k\left(\ln z+\psi \left(a+k\right)-\psi \left(1+k\right)-\psi \left(n+k+1\right)\right)+\frac{1}{\mathrm{\Gamma }\left(a\right)}\sum _{k=1}^n\frac{(k-1)!{\left(1-a+k\right)}_{n-k}}{(n-k)!}{z}^{-k}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.2(i), §33.6, §6.6, Erratum (V1.0.12) for Equations (13.2.9), (13.2.10)

Permalink:

http://dlmf.nist.gov/13.2.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.12):

The condition $a\ne 0,-1,-2,\mathrm{\dots }$ that appeared below this equation now appears ahead of the equation.

Suggested 2016-07-05 by Adri Olde Daalhuis

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

… ►Except when $a=0,-1,\mathrm{\dots }$ (polynomial cases), …

7: 12.11 Zeros

… ►Lastly, when $a=-n-\frac{1}{2}$, $n=1,2,\mathrm{\dots }$ (Hermite polynomial case) $U(a,x)$ has $n$ zeros and they lie in the interval $[-2\sqrt{-a},2\sqrt{-a}]$. …

8: 18.21 Hahn Class: Interrelations

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§18.21(ii) Limit Relations and Special Cases

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Figure 18.21.1: Askey scheme. …(This is with the convention that the real and imaginary parts of the parameters are counted separately in the case of the continuous Hahn polynomials.) Magnify

9: 18.33 Polynomials Orthogonal on the Unit Circle

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§18.33(iv) Special Cases

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10: 13.14 Definitions and Basic Properties

… ►Except when $\mu -\kappa =-\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$ (polynomial cases), …