# integer degree and order

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##### 1: 30.8 Expansions in Series of Ferrers Functions
30.8.7 $\frac{k^{2}a^{m}_{n,k}(\gamma^{2})}{a^{m}_{n,k-1}(\gamma^{2})}=\frac{\gamma^{2% }}{16}+O\left(\frac{1}{k}\right),$
30.8.8 $\frac{\lambda^{m}_{n}\left(\gamma^{2}\right)-B_{k}}{A_{k}}\frac{a^{m}_{n,k}(% \gamma^{2})}{a^{m}_{n,k-1}(\gamma^{2})}=1+O\left(\frac{1}{k^{4}}\right).$
##### 2: 30.1 Special Notation
 $x$ real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, $-1. … order, a nonnegative integer. degree, an integer $n=m,m+1,m+2,\dots$. integer. …
##### 3: 30.16 Methods of Computation
30.16.4 $\alpha_{p,d}-\lambda^{m}_{n}\left(\gamma^{2}\right)=O\left(\frac{\gamma^{4d}}{% 4^{2d+1}((m+2d-1)!(m+2d+1)!)^{2}}\right),$ $d\to\infty$.
##### 5: 30.11 Radial Spheroidal Wave Functions
30.11.6 $S^{m(j)}_{n}\left(z,\gamma\right)=\begin{cases}\psi_{n}^{(j)}(\gamma z)+O\left% (z^{-2}e^{|\Im z|}\right),&j=1,2,\\ \psi_{n}^{(j)}(\gamma z)\left(1+O\left(z^{-1}\right)\right),&j=3,4.\end{cases}$
##### 6: 30.3 Eigenvalues
30.3.2 $\lambda^{m}_{n}\left(\gamma^{2}\right)=n(n+1)-\tfrac{1}{2}\gamma^{2}+O\left(n^% {-2}\right),$ $n\to\infty$,
##### 7: 14.21 Definitions and Basic Properties
14.21.1 $\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+\left(\nu(\nu+1)-\frac{\mu^{2}}{1-z^{2}}\right)w=0.$
##### 8: Bibliography S
• R. Szmytkowski (2009) On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 46 (1), pp. 231–260.
• R. Szmytkowski (2011) On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Chem. 49 (7), pp. 1436–1477.
• R. Szmytkowski (2012) On parameter derivatives of the associated Legendre function of the first kind (with applications to the construction of the associated Legendre function of the second kind of integer degree and order). J. Math. Anal. Appl. 386 (1), pp. 332–342.
• ##### 9: 10.19 Asymptotic Expansions for Large Order
10.19.9 $\rselection{{H^{(1)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)\\ {H^{(2)}_{\nu}}\left(\nu+a\nu^{\frac{1}{3}}\right)}\sim\frac{2^{\frac{4}{3}}}{% \nu^{\frac{1}{3}}}e^{\mp\pi i/3}\operatorname{Ai}\left(e^{\mp\pi i/3}2^{\frac{% 1}{3}}a\right)\sum_{k=0}^{\infty}\frac{P_{k}(a)}{\nu^{2k/3}}+\frac{2^{\frac{5}% {3}}}{\nu}e^{\pm\pi i/3}\operatorname{Ai}'\left(e^{\mp\pi i/3}2^{\frac{1}{3}}a% \right)\sum_{k=0}^{\infty}\frac{Q_{k}(a)}{\nu^{2k/3}},$
##### 10: 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). …