# integer order

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##### 2: 14.6 Integer Order
###### §14.6(ii) Negative IntegerOrders
For connections between positive and negative integer orders see (14.9.3), (14.9.4), and (14.9.13). …
##### 3: 11.1 Special Notation
 $x$ real variable. … integer order. …
##### 5: 14.21 Definitions and Basic Properties
14.21.1 $\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$
###### §14.21(iii) Properties
This includes, for example, the Wronskian relations (14.2.7)–(14.2.11); hypergeometric representations (14.3.6)–(14.3.10) and (14.3.15)–(14.3.20); results for integer orders (14.6.3)–(14.6.5), (14.6.7), (14.6.8), (14.7.6), (14.7.7), and (14.7.11)–(14.7.16); behavior at singularities (14.8.7)–(14.8.16); connection formulas (14.9.11)–(14.9.16); recurrence relations (14.10.3)–(14.10.7). …
##### 6: 11.4 Basic Properties
###### §11.4(i) Half-IntegerOrders
11.4.1 $\mathbf{K}_{n+\frac{1}{2}}\left(z\right)=\left(\frac{2}{\pi z}\right)^{\frac{1% }{2}}\sum_{m=0}^{n}\frac{(2m)!\,2^{-2m}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{n-2m},$
11.4.2 $\mathbf{L}_{n+\frac{1}{2}}\left(z\right)=I_{-n-\frac{1}{2}}\left(z\right)-% \left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\sum_{m=0}^{n}\frac{(-1)^{m}(2m)!\,2% ^{-2m}}{m!\,(n-m)!}\,(\tfrac{1}{2}z)^{n-2m},$
##### 7: 14.34 Software
• Olver and Smith (1983). Integer order. Fortran.

• ##### 8: Philip J. Davis
Olver had been recruited to write the Chapter “Bessel Functions of Integer Order” for A&S by Milton Abramowitz, who passed away suddenly in 1958. …
##### 9: Frank W. J. Olver
Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. …
##### 10: Bibliography
• A. Adelberg (1996) Congruences of $p$-adic integer order Bernoulli numbers. J. Number Theory 59 (2), pp. 374–388.
• F. A. Alhargan (2000) Algorithm 804: Subroutines for the computation of Mathieu functions of integer orders. ACM Trans. Math. Software 26 (3), pp. 408–414.
• R. W. B. Ardill and K. J. M. Moriarty (1978) Spherical Bessel functions $j_{n}$ and $y_{n}$ of integer order and real argument. Comput. Phys. Comm. 14 (3-4), pp. 261–265.