Appendix/SpecialFunctions

Special Functions

Gamma Function

Gamma Function
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Template_Name	Resulting Equation
EQ_Gamma01	$\mathrm{\Gamma }(z)\mathrm{\Gamma }(1-z)$ $=$ $\frac{\pi }{\sin (\pi z)}$ $|$ for example, if $z\to (m-n+\frac{1}{2})$ $\Rightarrow \mathrm{\Gamma }(m-n+\frac{1}{2})\mathrm{\Gamma }(n-m+\frac{1}{2})$ $=$ $\pi \{\sin [\frac{\pi }{2}+\pi (m-n)]{\}}^{-1}$   $=$ $\pi (-1{)}^{m-n}$ DLMF §5.5(ii) $|$ Valid for:    $z\ne 0,\pm 1,\pm 2,$ … $|$	A. Erdélyi (1953):  Volume I, §1.2, p. 3, eq. (6) Wikipedia

Complete Elliptic Integrals

Complete Elliptic Integral …
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EQ_EllipticIntegral01	… of the First Kind $\frac{2K(k)}{\pi }$ $=$ $1+(\frac{1}{2}{)}^2{k}^2+(\frac{1\cdot 3}{2\cdot 4}{)}^2{k}^4+(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}{)}^2{k}^6+(\frac{1\cdot 3\cdot 5\cdot 7}{2^7\cdot 3}{)}^2{k}^8+\cdots +[\frac{(2n-1)!!}{2^{n}n!}{]}^2{k}^{2n}+\cdots$ Gradshteyn & Ryzhik (1965), §8.113.1	DLMF §19.5.1 Wolfram's Mathworld Wikipedia
EQ_EllipticIntegral03	… of the First Kind (alternate expression) $K(\mu )$ $=$ $\ln \frac{4}{k^{\text{'}}}+\frac{1}{2^2}(\ln \frac{4}{k^{\text{'}}}-\frac{2}{1\cdot 2})(k^{\prime }{)}^2+(\frac{1\cdot 3}{2\cdot 4}{)}^2(\ln \frac{4}{k^{\text{'}}}-\frac{2}{1\cdot 2}-\frac{2}{3\cdot 4})(k^{\prime }{)}^4+(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}{)}^2(\ln \frac{4}{k^{\text{'}}}-\frac{2}{1\cdot 2}-\frac{2}{3\cdot 4}-\frac{2}{5\cdot 6})(k^{\prime }{)}^6+\cdots$ Gradshteyn & Ryzhik (1965), §8.113.3 where:   $k^{\text{'}}\equiv (1-{\mu }^2{)}^{1/2}$
EQ_EllipticIntegral02	… of the Second Kind $\frac{2E(k)}{\pi }$ $=$ $1-\frac{1}{2^2}{k}^2-\frac{1^2\cdot 3}{2^2\cdot 4^2}{k}^4-(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}{)}^2\frac{k^6}{5}-(\frac{1\cdot 3\cdot 5\cdot 7}{2^7\cdot 3}{)}^2\frac{k^8}{7}-\cdots -[\frac{(2n-1)!!}{2^{n}n!}{]}^2\frac{k^{2n}}{2n-1}-\cdots$ Gradshteyn & Ryzhik (1965), §8.114.1	DLMF §19.5.2 Wolfram's MathWorld Wikipedia
EQ_EllipticIntegral04	… of the Second Kind (alternate expression) $E(\mu )$ $=$ $1+\frac{1}{2}(\ln \frac{4}{k^{\prime }}-\frac{1}{1\cdot 2})(k^{\prime }{)}^2+\frac{1^2\cdot 3}{2^2\cdot 4}(\ln \frac{4}{k^{\prime }}-\frac{2}{1\cdot 2}-\frac{1}{3\cdot 4})(k^{\prime }{)}^4+\frac{1^2\cdot 3^2\cdot 5}{2^2\cdot 4^2\cdot 6}(\ln \frac{4}{k^{\prime }}-\frac{2}{1\cdot 2}-\frac{2}{3\cdot 4}-\frac{1}{5\cdot 6})(k^{\prime }{)}^6+\cdots$ Gradshteyn & Ryzhik (1965), §8.114.3 where:   $k^{\text{'}}\equiv (1-{\mu }^2{)}^{1/2}$

See also:

• W. J. Cody (1965, Mathematics of Computation, Vol. 19, No. 89, pp. 105 - 112), "Chebyshev Approximations for the Complete Elliptic Integrals K and E".
• "Chebyshev Polynomial Expansions of Complete Elliptic Integrals," by W. J. Cody (Argonne National Laboratory)

Toroidal Function Evaluations

Analytic Expressions & Plots

Toroidal Function Evaluations
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EQ_PminusHalf01	$P_{-\frac{1}{2}}(z)$ $=$ $\frac{2}{\pi }\left[\frac{2}{z+1}{]}^{1/2}K\right(\sqrt{\frac{z-1}{z+1}})$       for example … $P_{-\frac{1}{2}}(\cosh \eta )$ $=$ $[\frac{\pi }{2}\cdot \cosh \frac{\eta }{2}{]}^{-1}K(\tanh \frac{\eta }{2})$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.1) Abramowitz & Stegun (1995), p. 337, eq. (8.13.2) NOTE: We have explicitly demonstrated that an alternate, equivalent expression is: $P_{-\frac{1}{2}}(\cosh \eta )$ $=$ $\frac{\sqrt{2}}{\pi }(\sinh \eta {)}^{-1/2}kK(k)$       where:     $k$ $\equiv$ $[2/(\coth \eta +1){]}^{1/2}.$	P0minus1Half
EQ_QminusHalf01	$Q_{-\frac{1}{2}}(z)$ $=$ $\sqrt{\frac{2}{z+1}}K\left(\sqrt{\frac{2}{z+1}}\right)$       for example … $Q_{-\frac{1}{2}}(\cosh \eta )$ $=$ $2e^{-\eta /2}K(e^{-\eta })$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.3) Abramowitz & Stegun (1995), p. 337, eq. (8.13.4)	Q0minusHalf
EQ_PplusHalf01	$P_{+\frac{1}{2}}(z)$ $=$ $\frac{2}{\pi }[z+\sqrt{z^2-1}{]}^{1/2}E(\sqrt{\frac{2(z^2-1{)}^{1/2}}{z+(z^2-1{)}^{1/2}}})$       for example … $P_{+\frac{1}{2}}(\cosh \eta )$ $=$ $\frac{2}{\pi }{e}^{\eta /2}E(\sqrt{1-e^{-2\eta }})$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.5) Abramowitz & Stegun (1995), p. 337, eq. (8.13.6) NOTE: It appears as though an alternate, equivalent expression is: $P_{+\frac{1}{2}}(\cosh \eta )$ $=$ $\frac{\sqrt{2}}{\pi }(\sinh \eta {)}^{+1/2}{k}^{-1}E(k)$       where:     $k$ $\equiv$ $[2/(\coth \eta +1){]}^{1/2}.$	P0plusHalf
EQ_QplusHalf01	$Q_{+\frac{1}{2}}(z)$ $=$ $z\sqrt{\frac{2}{z+1}}K\left(\sqrt{\frac{2}{z+1}}\right)-[2(z+1){]}^{1/2}E(\sqrt{\frac{2}{z+1}})$ Abramowitz & Stegun (1995), p. 337, eq. (8.13.7)	Q0plusHalf
EQ_Q1minusHalf01	$Q_{-\frac{1}{2}}^1(z)$ $=$ $-[\frac{1}{2(z-1)}{]}^{1/2}E(k)$ where:   $k=\sqrt{\frac{2}{z+1}}.$ (see our associated derivation)	ABSQ1minusHalf
EQ_Q2minusHalf01	$Q_{-\frac{1}{2}}^2(z)$ $=$ $\frac{4zE(k)-(z-1)K(k)}{[2^3(z+1)(z-1{)}^2{]}^{1/2}}$ where:   $k\equiv \sqrt{\frac{2}{z+1}}.$ (see our associated derivation)	Q2minusHalf

Caption for Plots

Caption for Plots:   Here we explain how we assembled the various plots — shown immediately above in the right-hand column of the "Toroidal Function Evaluations" table — that depict the behavior of various associated Legendre (toroidal) functions (see the related discussion) having varying half-integer degrees $P_{-\frac{1}{2}}^0$, $P_{+\frac{1}{2}}^0$, $Q_{-\frac{1}{2}}^0$, $Q_{+\frac{1}{2}}^0$, $Q_{+\frac{3}{2}}^0,$ and (in association with a separate related discussion) having varying order $Q_{-\frac{1}{2}}^1$, $Q_{-\frac{1}{2}}^2$. For each choice of the integer indexes, $n\ge 0$ and $m\ge 0$, the relevant plot shows how the function, $X_{m-\frac{1}{2}}^n(z)$, varies with $z$. (Click on the small plot image to view an enlarged image.) In each plot … The solid green circular markers identify data that has been pulled directly from Table IX (p. 1923) of [MF53]; The solid orange circular markers identify function values that we have calculated using the relevant formulae as expressed herein in terms of the complete elliptic integrals, $K(k)$ and $E(k)$, where the relevant values of the elliptic integrals have been pulled directly from tabulated values published in pp. 535 - 537 of [CRC]. (See an accompanying sample of elliptic integral values extracted from [CRC].) The dashed red curve was also derived using formulae expressed in terms of the complete elliptic integrals, but the values of the elliptic integrals have been calculated using (double-precision versions of) algorithms drawn from Numerical Recipes. NOTE: The tabulated values of the function, $Q_{-\frac{1}{2}}^1$, that appear in Table IX (p. 1923) of [MF53] — also see immediately below — are all positive, whereas, according to our derivation, they should all be negative. Therefore, for comparison purposes of this specific function — both here and in our accompanying discussion — we have plotted the absolute value of the function, $|Q_{-\frac{1}{2}}^1(z)|$. ADDITIONAL NOTE:   In Example 4 on p. 340 of Abramowitz & Stegun (1995), we can pull one additional data point for comparison; specifically, they provide a high-precision evaluation of $Q_{-\frac{1}{2}}^0(z=2.6)=1.419337751$. As can be seen in the table of function values immediately below, this is entirely consistent with the lower-precision value that we have extracted from [MF53], and exactly matches the double-precision value we have calculated based on the Numerical Recipes algorithm.

Example Recurrence Relations

The above Toroidal Function Evaluations table provides analytic expressions for the pair of foundation functions, $P_{-\frac{1}{2}}^0(z)$ and $P_{+\frac{1}{2}}^0(z)$, and the associated pair of foundation functions, $Q_{-\frac{1}{2}}^0(z)$ and $Q_{+\frac{1}{2}}^0(z)$. From either pair of foundation functions, expressions for all other zero-order, half-integer degree toroidal functions can be obtained using a relatively simple recurrence relation drawn from the "Key Equation,"

	$(\nu -\mu +1)P_{\nu +1}^{\mu }(z)$	$=$	$(2\nu +1)z{P}_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)$
Abramowitz & Stegun (1995), p. 334, eq. (8.5.3)

NOTE: $Q_{\nu }^{\mu }$, as well as $P_{\nu }^{\mu }$, satisfies this same recurrence relation.

Specifically, letting $\mu \to 0$ and $\nu \to (m-\frac{1}{2})$, for all $m\ge 2$, we have,

$P{0}_{m-\frac{1}{2}}(z)$	$=$	$4\left[\frac{m-1}{2m-1}\right]z{P}_{m-\frac{3}{2}}^0(z)-\left[\frac{2m-3}{2m-1}\right]P_{m-\frac{5}{2}}^0(z);$     and,
$Q_{m-\frac{1}{2}}^0(z)$	$=$	$4\left[\frac{m-1}{2m-1}\right]z{Q}_{m-\frac{3}{2}}^0(z)-\left[\frac{2m-3}{2m-1}\right]Q_{m-\frac{5}{2}}^0(z).$

As examples, these two relations have been used to generate columns of numbers in the comparison table shown below for, respectively, the toroidal functions, $P_{+\frac{3}{2}}^0(z)$ and $Q_{+\frac{3}{2}}^0(z)$. For order-1 and order-2 toroidal functions, the above table provides analytic expressions only for (the functions of the lowest half-integer degree) $Q_{-\frac{1}{2}}^1(z)$ and $Q_{-\frac{1}{2}}^2(z)$. But, as we have detailed in an accompanying discussion, additional order-1 and order-2 expressions can be straightforwardly derived by drawing upon another key recurrence relation, namely,

	$P_{\nu }^{\mu +1}(z)$	$=$	$(z^2-1{)}^{-\frac{1}{2}}\{(\nu -\mu )z{P}_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)\}$
Abramowitz & Stegun (1995), p. 333, eq. (8.5.1)

NOTE: $Q_{\nu }^{\mu }$, as well as $P_{\nu }^{\mu }$, satisfies this same recurrence relation.

Specifically, after adopting the association, $\nu \to (n-\frac{1}{2})$, we have, when $\mu =0$,

$Q_{n-\frac{1}{2}}^1(z)$

$=$

$(n-\frac{1}{2})(z^2-1{)}^{-\frac{1}{2}}[z{Q}_{n-\frac{1}{2}}(z)-Q_{n-\frac{3}{2}}(z)]$

…

for $n\ge 1,$

and, when $\mu =1$,

$Q_{n-\frac{1}{2}}^2(z)$

$=$

$(z^2-1{)}^{-\frac{1}{2}}\{(n-\frac{3}{2})z{Q}_{n-\frac{1}{2}}^1(z)-(n+\frac{1}{2})Q_{n-\frac{3}{2}}^1(z)\}$

…

for $n\ge 1.$

As an example, the first of these two relations has been used to generate a column of numbers in the comparison table shown below for the toroidal function, $Q_{+\frac{1}{2}}^1(z)$.

Comparison with Table IX from MF53

To facilitate copying & pasting for immediate use by other researchers, here we present in a tab-delimited, plain-text format the evaluation of nine separate toroidal functions: (Top half of table) $P_{-\frac{1}{2}}^0$, $P_{+\frac{1}{2}}^0$ and $P_{+\frac{3}{2}}^0$; (Bottom half of table) $Q_{-\frac{1}{2}}^0$, $Q_{-\frac{1}{2}}^1$, $Q_{-\frac{1}{2}}^2$, $Q_{+\frac{1}{2}}^0$, $Q_{+\frac{1}{2}}^1$ and $Q_{+\frac{3}{2}}^0$. Each function has been evaluated for approximately 23 different argument values in the range, $1.0\le z\le 9.0$, and, for each function, two columns of function values have been presented: (Left column) Low-precision evaluation extracted directly from Table IX (p. 1923) of [MF53]; (Right column) Our double-precision evaluation based on a set of Numerical Recipes algorithms. One exception: The value listed under the "MF53" column for the evaluation of $Q_{-\frac{1}{2}}^0(z=2.6)$ is the high-precision value published on p. 340 of Abramowitz & Stegun (1995); notice that our high-precision evaluation matches all ten digits of their published value.

Top half of Table IX (p. 1923) of [MF53]

z P0m1Half(z) P0p1Half(z) P0p3Half(z) MF53 Our Calc. MF53 Our Calc. MF53 Our Calc. 1.0 1.0000 1.0000 1.0000 1.2 0.9763 9.763155118E-01 1.0728 1.072784040E+00 1.3910 1.391015961E+00 1.4 0.9549 9.549467781E-01 1.1416 1.141585331E+00 1.8126 1.812643692E+00 1.6 0.9355 9.355074856E-01 1.2070 1.206963827E+00 2.2630 2.263020336E+00 1.8 0.9177 9.176991005E-01 1.2694 1.269362428E+00 2.7406 2.740570128E+00 2.0 0.9013 9.012862994E-01 1.3291 1.329138155E+00 3.2439 3.243939648E+00 2.2 0.8861 8.860804115E-01 1.3866 1.386583505E+00 3.7719 3.771951476E+00 2.4 0.8719 8.719279330E-01 1.4419 1.441941436E+00 4.3236 4.323569952E+00 2.6 0.8587 8.587023595E-01 1.4954 1.495416274E+00 4.8979 4.897875630E+00 2.8 0.8463 8.462982520E-01 1.5472 1.547181667E+00 5.4941 5.494045473E+00 3.0 0.8346 8.346268417E-01 1.5974 1.597386605E+00 6.1113 6.111337473E+00 3.5 0.8082 8.081851582E-01 1.7169 1.716877977E+00 7.7427 7.742702172E+00 4.0 0.7850 7.849616703E-01 1.8290 1.828992729E+00 9.4930 9.492973996E+00 4.5 0.7643 7.643076802E-01 1.9349 1.934919997E+00 11.3555 1.135475076E+01 5.0 0.7457 7.457491873E-01 2.0356 2.035563839E+00 13.3220 1.332184253E+01 5.5 0.7289 7.289297782E-01 2.1316 2.131629923E+00 15.3890 1.538897617E+01 6.0 0.7136 7.135750093E-01 2.2237 2.223681177E+00 17.5520 1.755159108E+01 6.5 0.6995 6.994692725E-01 2.3122 2.312174942E+00 19.8060 1.980569307E+01 7.0 0.6864 6.864402503E-01 2.3975 2.397488600E+00 22.1480 2.214774685E+01 7.5 0.6743 6.743481630E-01 2.4799 2.479937758E+00 24.5750 2.457459486E+01 8.0 0.6631 6.630781433E-01 2.5598 2.559789460E+00 27.0830 2.708339486E+01 8.5 6.525347093E-01 2.637271986E+00 2.967157094E+01 9.0 6.426376817E-01 2.712582261E+00 3.233677457E+01

Bottom half of Table IX (p. 1923) of [MF53] ATTENTION:   Widen your browser window, or "zoom out," in order to obtain a proper view of the space-delimited columns of numbers in this table.

z Q0m1Half(z) Q1m1Half(z) Q2m1Half(z) Q0p1Half(z) Q1p1Half(z) Q0p3Half(z) MF53 Our Cal. MF53 Our Calc. MF53 Our Calc. M53 Our Calc. MF53 Our Calc. MF53 Our Calc. 1.1 2.8612 2.861192872E+00 2.3661 -2.366084077E+00 10.6440 1.064378304E+01 0.9788 9.787602829E-01 1.9471 -1.947110839E+00 0.4818 4.817841242E-01 1.2 2.5010 2.500956508E+00 1.7349 -1.734890983E+00 5.6518 5.651832631E+00 0.6996 6.995548314E-01 1.2524 -1.252395745E+00 0.2856 2.856355610E-01 1.4 2.1366 2.136571733E+00 1.2918 -1.291802851E+00 3.1575 3.157491205E+00 0.4598 4.597941602E-01 0.7618 -7.618218821E-01 0.14609 1.460918547E-01 1.6 1.9229 1.922920866E+00 1.0943 -1.094337965E+00 2.3230 2.323018870E+00 0.3430 3.430180260E-01 0.5501 -5.500770475E-01 0.09080 9.079816684E-02 1.8 1.7723 1.772268479E+00 0.9748 -9.748497733E-01 1.9018 1.901788930E+00 0.2720 2.720401772E-01 0.4285 -4.284853031E-01 0.06214 6.214026586E-02 2.0 1.6566 1.656638170E+00 0.8918 -8.917931374E-01 1.6454 1.645348489E+00 0.2240 2.240142929E-01 0.3489 -3.488955345E-01 0.04516 4.515872426E-02 2.2 1.5634 1.563378886E+00 0.8293 -8.292825549E-01 1.4712 1.471197798E+00 0.18932 1.893229696E-01 0.29263 -2.926294028E-01 0.03422 3.422108228E-02 2.4 1.4856 1.485653983E+00 0.7798 -7.797558474E-01 1.3441 1.344108936E+00 0.16312 1.631167365E-01 0.25076 -2.507568731E-01 0.02676 2.675556229E-02 2.6 1.419337751 1.419337751E+00 0.7391 -7.390875295E-01 1.2465 1.246521876E+00 0.14266 1.426580119E-01 0.21842 -2.184222751E-01 0.02143 2.143519083E-02 2.8 1.3617 1.361744950E+00 0.7048 -7.048053314E-01 1.1687 1.168702464E+00 0.12628 1.262756033E-01 0.19274 -1.927423405E-01 0.01751 1.751393553E-02 3.0 1.3110 1.311028777E+00 0.6753 -6.753219405E-01 1.1048 1.104816977E+00 0.11289 1.128885424E-01 0.17189 -1.718911443E-01 0.01454 1.454457729E-02 3.5 1.2064 1.206444997E+00 0.6163 -6.163068170E-01 0.9846 9.846190928E-01 0.08824 8.824567577E-02 0.13380 -1.338040913E-01 0.00966 9.664821286E-03 4.0 1.1242 1.124201960E+00 0.5713 -5.712994484E-01 0.8990 8.990205764E-01 0.07154 7.154134054E-02 0.10819 -1.081900595E-01 0.00682 6.819829619E-03 4.5 1.0572 1.057164923E+00 0.5353 -5.353494651E-01 0.8339 8.338659751E-01 0.05957 5.956966068E-02 0.08993 -8.992645608E-02 0.00503 5.029656514E-03 5.0 1.0011 1.001077380E+00 0.5057 -5.056928088E-01 0.7820 7.819717783E-01 0.05063 5.062950976E-02 0.07634 -7.633526879E-02 0.00384 3.837604899E-03 5.5 0.9532 9.532056775E-01 0.4806 -4.806378723E-01 0.7393 7.392682950E-01 0.04374 4.373774515E-02 0.06588 -6.588433822E-02 0.00301 3.008238619E-03 6.0 0.9117 9.116962715E-01 0.4591 -4.590784065E-01 0.7033 7.032568965E-01 0.03829 3.828867029E-02 0.05764 -5.763649873E-02 0.00241 2.410605139E-03 6.5 0.87524 8.752387206E-01 0.44025 -4.402537373E-01 0.67231 6.723067009E-01 0.03389 3.389003482E-02 0.05099 -5.098806037E-02 0.00197 1.967394932E-03 7.0 0.84288 8.428751774E-01 0.42362 -4.236198508E-01 0.64530 6.453008278E-01 0.03028 3.027740449E-02 0.04553 -4.553369214E-02 0.00163 1.630716095E-03 7.5 0.81389 8.138862008E-01 0.40877 -4.087751846E-01 0.62144 6.214442864E-01 0.02727 2.726650960E-02 0.04099 -4.099183107E-02 0.00137 1.369695722E-03 8.0 0.78772 7.877190099E-01 0.39542 -3.954155185E-01 0.60015 6.001530105E-01 0.02473 2.472532098E-02 0.03716 -3.716124286E-02 0.00116 1.163753807E-03 8.5 7.639406230E-01 -3.833053056E-01 5.809864341E-01 2.255696890E-02 -3.389458114E-02 9.987731857E-04 9.0 7.422062367E-01 -3.722587645E-01 5.636047532E-01 2.068890884E-02 -3.108168349E-02 8.648271474E-04

Relationships Between Various Associated Legendre Functions

Relationships Between Various Associated Legendre Functions
To insert a given equation into any Wiki document, type ... {{ Math/Template_Name }}		See also …
Template_Name	Resulting Equation
EQ_Toroidal00	$P_{\nu }^n(z)$ $=$ $\frac{\mathrm{\Gamma }(\nu +n+1)}{\mathrm{\Gamma }(\nu -n+1)}{P}_{\nu }^{-n}(z)$ A. Erdélyi (1953):  Volume I, §3.3.1, p. 140, eq. (7)
EQ_Toroidal01	$Q_{\nu }[t{t}^{\text{'}}-(t^2-1{)}^{1/2}(t^{\text{'}2}-1{)}^{1/2}\cos \psi ]$ $=$ $Q_{\nu }(t)P_{\nu }(t^{\text{'}})+2\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{Q}_{\nu }^n(t)P_{\nu }^{-n}(t^{\text{'}})\cos (n\psi )$ A. Erdélyi (1953):  Volume I, §3.11, p. 169, eq. (4) Valid for:     $t,t^{\text{'}}$  real         $1<t^{\text{'}}<t$         $\nu \ne -1,-2,-3,$ …         $\psi$   real	T. G. Cowling (1940):   p. 223 (note sign discrepancy in argument of $Q_{\nu }$) DLMF §14.18.5
EQ_Toroidal02	$Q_{n-1/2}^m(\lambda )$ $=$ $(-1{)}^n\frac{{\pi }^{3/2}}{\sqrt{2}\mathrm{\Gamma }(n-m+1/2)}(x^2-1{)}^{1/4}{P}_{m-1/2}^n(x),$ Gil, Segura, & Temme (2000):  eq. (8) where:     $\lambda \equiv x/\sqrt{x^2-1}$	Cohl et al. (2000), eq. (34) DLMF §14.19.v together with DLMF §14.3.10
EQ_Toroidal03	$Q_{\nu }^{\mu }(z)$ $=$ $e^{i\mu \pi }(2\pi {)}^{-\frac{1}{2}}(z^2-1{)}^{\mu /2}\mathrm{\Gamma }(\mu +\frac{1}{2})\{\underset{0}{\overset{\pi }{\int }}(z-\cos t{)}^{-\mu -\frac{1}{2}}\cos [(\nu +\frac{1}{2})t]dt-\cos (\nu \pi )\underset{0}{\overset{\mathrm{\infty }}{\int }}(z+\cosh t{)}^{-\mu -\frac{1}{2}}{e}^{-(\nu +\frac{1}{2})t}dt\}$ A. Erdélyi (1953):  Volume I, §3.7, p. 156, eq. (10) Valid for:     $\mathrm{R}\mathrm{e}\nu >-\frac{1}{2}$      and     $\mathrm{R}\mathrm{e}(\nu +\mu +1)>0.$
EQ_Toroidal04	$(\nu -\mu +1)P_{\nu +1}^{\mu }(z)$ $=$ $(2\nu +1)z{P}_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)$ Abramowitz & Stegun (1995), p. 334, eq. (8.5.3) NOTE: $Q_{\nu }^{\mu }$, as well as $P_{\nu }^{\mu }$, satisfies this same recurrence relation.	Cohl & Tohline (1999), §2.2.2, eq. (25) Guatschi (1965), p. 490, procedure toroidal
EQ_Toroidal05	$\underset{a}{\overset{b}{\int }}[(\nu -\sigma )(\nu +\sigma +1)+({\rho }^2-{\mu }^2)(1-z^2{)}^{-1}]w_{\nu }^{\mu }{w}_{\sigma }^{\rho }dz$ $=$ $[z(\nu -\sigma )w_{\nu }^{\mu }{w}_{\sigma }^{\rho }+(\sigma +\rho )w_{\nu }^{\mu }{w}_{\sigma -1}^{\rho }-(\nu +\mu )w_{\nu -1}^{\mu }{w}_{\sigma }^{\rho }{]}_a^b$ A. Erdélyi (1953):  Volume I, §3.12, p. 169, eq. (1) where, $w_{\nu }^{\mu }(z)$ and $w_{\sigma }^{\rho }(z)$ denote any solutions of Legendre's differential equation
EQ_Toroidal06	$(\xi -z)\sum _{m=0}^n(2m+1)P_m(z)Q_m(\xi )$ $=$ $1-(\ell +1)[P_{\ell +1}(z)Q_{\ell }(\xi )-P_{\ell }(z)Q_{\ell +1}(\xi )]$ Abramowitz & Stegun (1995), p. 335, eq. (8.9.2)	Erdélyi (1953):  Volume I, §3.8, p. 162, eq. (21)
EQ_Toroidal07	$P_{\nu }^{\mu +1}(z)$ $=$ $(z^2-1{)}^{-\frac{1}{2}}\{(\nu -\mu )z{P}_{\nu }^{\mu }(z)-(\nu +\mu )P_{\nu -1}^{\mu }(z)\}$ Abramowitz & Stegun (1995), p. 333, eq. (8.5.1) NOTE: $Q_{\nu }^{\mu }$, as well as $P_{\nu }^{\mu }$, satisfies this same recurrence relation.
EQ_Toroidal08	$Q_{-\frac{1}{2}}^{\mu }(z)+2\sum _{n=1}^{\mathrm{\infty }}{Q}_{n-\frac{1}{2}}^{\mu }(z)\cos (n\nu )$ $=$ $e^{i\mu \pi }(\frac{\pi }{2}{)}^{1/2}\mathrm{\Gamma }(\mu +\frac{1}{2})[\frac{(z^2-1{)}^{\mu /2}}{(z-\cos \nu {)}^{\mu +\frac{1}{2}}}]$ A. Erdélyi (1953):  Volume I, §3.10, p. 166, eq. (3) Valid for:     $\mathrm{R}\mathrm{e}\mu >-\frac{1}{2}$