# of arbitrary order

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##### 11: 14.24 Analytic Continuation
14.24.1 $P^{-\mu}_{\nu}\left(ze^{s\pi i}\right)=e^{s\nu\pi i}P^{-\mu}_{\nu}\left(z% \right)+\frac{2i\sin\left(\left(\nu+\frac{1}{2}\right)s\pi\right)e^{-s\pi i/2}% }{\cos\left(\nu\pi\right)\Gamma\left(\mu-\nu\right)}\boldsymbol{Q}^{\mu}_{\nu}% \left(z\right),$
14.24.2 $\boldsymbol{Q}^{\mu}_{\nu}\left(ze^{s\pi i}\right)=(-1)^{s}e^{-s\nu\pi i}% \boldsymbol{Q}^{\mu}_{\nu}\left(z\right),$
14.24.3 $P^{-\mu}_{\nu,s}\left(z\right)=e^{s\mu\pi i}P^{-\mu}_{\nu}\left(z\right),$
14.24.4 $\boldsymbol{Q}^{\mu}_{\nu,s}\left(z\right)=e^{-s\mu\pi i}\boldsymbol{Q}^{\mu}_% {\nu}\left(z\right)-\frac{\pi i\sin\left(s\mu\pi\right)}{\sin\left(\mu\pi% \right)\Gamma\left(\nu-\mu+1\right)}P^{-\mu}_{\nu}\left(z\right),$
##### 12: 11.1 Special Notation
###### §11.1 Special Notation
 $x$ real variable. … real or complex order. integer order. … arbitrary small positive constant.
##### 13: 10.1 Special Notation
 $m,n$ integers. In §§10.47–10.71 $n$ is nonnegative. … arbitrary small positive constant. …
For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).
##### 14: Bibliography O
• F. W. J. Olver and F. Stenger (1965) Error bounds for asymptotic solutions of second-order differential equations having an irregular singularity of arbitrary rank. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 244–249.
• F. W. J. Olver (1977b) Connection formulas for second-order differential equations having an arbitrary number of turning points of arbitrary multiplicities. SIAM J. Math. Anal. 8 (4), pp. 673–700.
• ##### 15: 11.11 Asymptotic Expansions of Anger–Weber Functions
11.11.8 $\mathbf{A}_{\nu}\left(\lambda\nu\right)\sim\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$ ($<\pi$),
##### 16: 3.6 Linear Difference Equations
3.6.9 $\left|\frac{e_{N}}{p_{N}p_{N+1}}\right|\leq\epsilon\min_{1\leq n\leq M}\left|% \frac{e_{n}}{p_{n}p_{n+1}}\right|.$
##### 17: 24.16 Generalizations
In no particular order, other generalizations include: Bernoulli numbers and polynomials with arbitrary complex index (Butzer et al. (1992)); Euler numbers and polynomials with arbitrary complex index (Butzer et al. (1994)); q-analogs (Carlitz (1954a), Andrews and Foata (1980)); conjugate Bernoulli and Euler polynomials (Hauss (1997, 1998)); Bernoulli–Hurwitz numbers (Katz (1975)); poly-Bernoulli numbers (Kaneko (1997)); Universal Bernoulli numbers (Clarke (1989)); $p$-adic integer order Bernoulli numbers (Adelberg (1996)); $p$-adic $q$-Bernoulli numbers (Kim and Kim (1999)); periodic Bernoulli numbers (Berndt (1975b)); cotangent numbers (Girstmair (1990b)); Bernoulli–Carlitz numbers (Goss (1978)); Bernoulli-Padé numbers (Dilcher (2002)); Bernoulli numbers belonging to periodic functions (Urbanowicz (1988)); cyclotomic Bernoulli numbers (Girstmair (1990a)); modified Bernoulli numbers (Zagier (1998)); higher-order Bernoulli and Euler polynomials with multiple parameters (Erdélyi et al. (1953a, §§1.13.1, 1.14.1)).
##### 18: Bibliography M
• B. T. M. Murphy and A. D. Wood (1997) Hyperasymptotic solutions of second-order ordinary differential equations with a singularity of arbitrary integer rank. Methods Appl. Anal. 4 (3), pp. 250–260.
• ##### 19: 14.15 Uniform Asymptotic Approximations
###### §14.15(i) Large $\mu$, Fixed $\nu$
For the interval $-1 with fixed $\nu$, real $\mu$, and arbitrary fixed values of the nonnegative integer $J$, … In this and subsequent subsections $\delta$ denotes an arbitrary constant such that $0<\delta<1$. …
14.15.5 $\alpha=\frac{\nu+\frac{1}{2}}{\mu}\,(<1),$
For asymptotic expansions and explicit error bounds, see Dunster (2003b). …
##### 20: 14.1 Special Notation
###### §14.1 Special Notation
 $x$, $y$, $\tau$ real variables. … unless stated otherwise, nonnegative integers, used for order and degree, respectively. general order and degree, respectively. … arbitrary small positive constant. …