# Jordan function

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##### 1: 27.2 Functions
27.2.11 $J_{k}\left(n\right)=\sum_{\left(\left(d_{1},\dots,d_{k}\right),n\right)=1}1,$
This is Jordan’s function. …
##### 2: 27.6 Divisor Sums
27.6.7 $\sum_{d\mathbin{|}n}\mu\left(d\right)\left(\frac{n}{d}\right)^{k}=J_{k}\left(n% \right),$
##### 3: 5.16 Sums
###### §5.16 Sums
5.16.2 $\sum_{k=1}^{\infty}\frac{1}{k}\psi'\left(k+1\right)=\zeta\left(3\right)=-\frac% {1}{2}\psi''\left(1\right).$
For further sums involving the psi function see Hansen (1975, pp. 360–367). For sums of gamma functions see Andrews et al. (1999, Chapters 2 and 3) and §§15.2(i), 16.2. For related sums involving finite field analogs of the gamma and beta functions (Gauss and Jacobi sums) see Andrews et al. (1999, Chapter 1) and Terras (1999, pp. 90, 149).
##### 6: Peter L. Walker
Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. Walker’s books are An Introduction to Complex Analysis, published by Hilger in 1974, The Theory of Fourier Series and Integrals, published by Wiley in 1986, Elliptic Functions. A Constructive Approach, published by Wiley in 1996, and Examples and Theorems in Analysis, published by Springer in 2004. …
• ##### 7: 24.17 Mathematical Applications
Then with the notation of §24.2(iii)See Milne-Thomson (1933), Nörlund (1924), or Jordan (1965). …
###### §24.17(ii) Spline Functions
The functions
##### 8: Bibliography J
• D. L. Jagerman (1974) Some properties of the Erlang loss function. Bell System Tech. J. 53, pp. 525–551.
• D. S. Jones (2001) Asymptotics of the hypergeometric function. Math. Methods Appl. Sci. 24 (6), pp. 369–389.
• D. S. Jones (2006) Parabolic cylinder functions of large order. J. Comput. Appl. Math. 190 (1-2), pp. 453–469.
• C. Jordan (1939) Calculus of Finite Differences. Hungarian Agent Eggenberger Book-Shop, Budapest.
• C. Jordan (1965) Calculus of Finite Differences. 3rd edition, AMS Chelsea, Providence, RI.
• ##### 9: 26.8 Set Partitions: Stirling Numbers
###### §26.8(ii) Generating Functions
26.8.8 $\sum_{n=0}^{\infty}s\left(n,k\right)\frac{x^{n}}{n!}=\frac{(\ln\left(1+x\right% ))^{k}}{k!},$ $|x|<1$,
26.8.13 $\sum_{n,k=0}^{\infty}S\left(n,k\right)\frac{x^{n}}{n!}y^{k}=\exp\left(y({% \mathrm{e}}^{x}-1)\right).$
26.8.40 $s\left(n+1,k+1\right)\sim(-1)^{n-k}\frac{n!}{k!}(\gamma+\ln n)^{k},$ $n\to\infty$,
##### 10: 26.1 Special Notation
(For other notation see Notation for the Special Functions.) … The main functions treated in this chapter are:
 $\genfrac{(}{)}{0.0pt}{}{m}{n}$ binomial coefficient. … number of partitions of $n$. …
Other notations for $s\left(n,k\right)$, the Stirling numbers of the first kind, include $S_{n}^{(k)}$ (Abramowitz and Stegun (1964, Chapter 24), Fort (1948)), $S_{n}^{k}$ (Jordan (1939), Moser and Wyman (1958a)), $\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}$ (Milne-Thomson (1933)), $(-1)^{n-k}S_{1}(n-1,n-k)$ (Carlitz (1960), Gould (1960)), $(-1)^{n-k}\left[n\atop k\right]$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). Other notations for $S\left(n,k\right)$, the Stirling numbers of the second kind, include $\mathscr{S}^{(k)}_{n}$ (Fort (1948)), $\mathfrak{S}_{n}^{k}$ (Jordan (1939)), $\sigma_{n}^{k}$ (Moser and Wyman (1958b)), $\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}$ (Milne-Thomson (1933)), $S_{2}(k,n-k)$ (Carlitz (1960), Gould (1960)), $\left\{n\atop k\right\}$ (Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).