Notations P

$\mbox{P}_{\mbox{\scriptsize I}}$ , $\mbox{P}_{\mbox{\scriptsize II}}$ , $\mbox{P}_{\mbox{\scriptsize III}}$ , $\mbox{P}^{\prime}_{\mbox{\scriptsize III}}$ , $\mbox{P}_{\mbox{\scriptsize IV}}$ , $\mbox{P}_{\mbox{\scriptsize V}}$ , $\mbox{P}_{\mbox{\scriptsize VI}}$: Painlevé transcendents; §32.2(i)
$p_{\NVar{k}}\left(\NVar{n}\right)$: total number of partitions of $n$ into at most $k$ parts; §26.9(i)
$P(\NVar{z})=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$: alternative notation for the complementary error function; §7.1
(with $\operatorname{erfc}\NVar{z}$ : complementary error function)
$p_{\NVar{k}}\left(\NVar{n}\right)$: number of partitions of $n$ into at most $k$ parts; §26.9(i)
$P_{\NVar{n}}\left(\NVar{x}\right)$: Legendre polynomial; Table 18.3.1
$P^{*}_{\NVar{n}}\left(\NVar{x}\right)$: shifted Legendre polynomial; Table 18.3.1
$\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$: Ferrers function of the first kind; §14.2(ii)
(with $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind)
$P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$: Legendre function of the first kind; §14.2(ii)
(with $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind)
$P_{\NVar{z}}(\NVar{a})=\gamma\left(a,z\right)$: notation used by Batchelder (1967, p. 63); §8.1
(with $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function)
$\mathsf{P}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$: conical function; §14.20(i)
$P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x}\right)$: Jacobi polynomial; Table 18.3.1
$P_{\NVar{n}}^{(\NVar{\lambda})}(\NVar{x})=C^{(\lambda)}_{n}\left(x\right)$: notation used by Szegő (1975, §4.7); §18.1(iii)
(with $C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}\right)$ : ultraspherical (or Gegenbauer) polynomial)
$P^{(\NVar{\gamma},\NVar{\delta})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$: Jacobi function of matrix argument; (35.7.2)
$\mathrm{P}_{\NVar{\nu}}^{\NVar{\mu}}(\NVar{x})=\mathsf{P}^{\mu}_{\nu}\left(x\right)$: notation used by Erdélyi et al. (1953a), Olver (1997b); §14.1
(with $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind)
$\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$: Ferrers function of the first kind; (14.3.1)
$P_{\NVar{\nu}}^{\NVar{\mu}}(\NVar{x})=\mathsf{P}^{\mu}_{\nu}\left(x\right)$: notation used by Magnus et al. (1966); §14.1
(with $\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$ : Ferrers function of the first kind)
$P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$: associated Legendre function of the first kind; §14.21(i)
$\mathfrak{P}_{\NVar{\nu}}^{\NVar{\mu}}(\NVar{z})=P^{\mu}_{\nu}\left(z\right)$: notation used by Magnus et al. (1966); §14.1
(with $P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$ : associated Legendre function of the first kind)
$P\left(\NVar{a},\NVar{z}\right)$: normalized incomplete gamma function; (8.2.4)
$p\left(\NVar{\mathrm{condition}},\NVar{n}\right)$: restricted number of partitions of $n$ ; §26.10(i)
$\wp\left(\NVar{z}\right)$ (= $\wp\left(z|\mathbb{L}\right)$ = $\wp\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\wp$ -function; (23.2.4)
$p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ ; §26.9(i)
$p_{\NVar{k}}\left(\NVar{\mathcal{D}},\NVar{n}\right)$: number of partitions of $n$ into at most $k$ distinct parts; §26.10(i)
$P_{\NVar{\ell}}(\NVar{\epsilon},\NVar{r})=(2\ell+1)!f\left(\epsilon,\ell;r% \right)/2^{\ell+1}$: notation used by Curtis (1964a); item Curtis (1964a):
(with $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function and $!$ : factorial (as in $n!$ ))
$H_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Hermite polynomial; §18.30(iv)
$P_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Legendre polynomial; (18.30.6)
${\mathscr{P}}^{\NVar{\lambda}}_{\NVar{n}}\left(\NVar{x};\NVar{\phi},\NVar{c}\right)$: associated Meixner–Pollaczek polynomial; §18.30(v)
$P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$: Meixner–Pollaczek polynomial; §18.19
$P^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{m},\NVar{n}}\left(\NVar{x}% ,\NVar{y}\right)$: triangle polynomial; (18.37.7)
$P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$: associated Jacobi polynomial; (18.30.4)
$\Pi(\NVar{n};\NVar{\phi}\backslash\NVar{\alpha})=\Pi\left(\phi,\alpha^{2},k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind)
$\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$: Weierstrass $\wp$ -function; (23.3.8)
$P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b}\right)$: Pollaczek polynomial; (18.35.4)
$p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b};\NVar{q}\right)$: little $q$ -Jacobi polynomial; (18.27.13)
$P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(\NVar{x};\NVar{c},\NVar{d};% \NVar{q}\right)$: big $q$ -Jacobi polynomial; (18.27.6)
$p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{\overline{a}},\NVar{% \overline{b}}\right)$: continuous Hahn polynomial; §18.19
$P_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c};\NVar{q}\right)$: big $q$ -Jacobi polynomial; (18.27.5)
$p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,|\,\NVar{q}\right)$: Askey–Wilson polynomial; (18.28.1)
$P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$: Riemann’s $P$ -symbol for solutions of the generalized hypergeometric differential equation; (15.11.3)
$\operatorname{ph}$: phase; (1.9.7)
$\phi\left(\NVar{n}\right)$: Euler’s totient; (27.2.7)
$\phi\left(\NVar{z}\right)$: Airy phase function; (9.8.8)
$\Phi(\NVar{z})=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$: alternative notation for the complementary error function; §7.1
(with $\operatorname{erfc}\NVar{z}$ : complementary error function)
$\phi_{\NVar{k}}\left(\NVar{n}\right)$: sum of powers of integers relatively prime to a number; (27.2.6)
$\phi_{\NVar{\nu}}\left(\NVar{x}\right)$: phase of derivatives of Bessel functions; (10.18.3)
$\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda}}\left(\NVar{t}\right)$: Jacobi function; (15.9.11)
$\phi\left(\NVar{z},\NVar{s}\right)=\operatorname{Li}_{s}\left(z\right)$: notation used by (Truesdell, 1945); §25.12(ii)
(with $\operatorname{Li}_{\NVar{s}}\left(\NVar{z}\right)$ : polylogarithm)

$\varphi_{\NVar{n},\NVar{m}}\left(\NVar{z},\NVar{q}\right)$: combined theta function; §20.11(v)
$\Phi_{\NVar{K}}\left(\NVar{t};\NVar{\mathbf{x}}\right)$: cuspoid catastrophe of codimension $K$ ; (36.2.1)
$\Phi(\NVar{a};\NVar{b};\NVar{z})=M\left(a,b,z\right)$: notation used by Humbert (1920); §13.1
(with $M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : $={{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ Kummer confluent hypergeometric function)
$\phi\left(\NVar{\rho},\NVar{\beta};\NVar{z}\right)$: generalized Bessel function; (10.46.1)
$\Phi\left(\NVar{z},\NVar{s},\NVar{a}\right)$: Lerch’s transcendent; (25.14.1)
$\Phi^{(\mathrm{E})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$: elliptic umbilic catastrophe; (36.2.2)
$\Phi^{(\mathrm{H})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$: hyperbolic umbilic catastrophe; (36.2.3)
$\Phi^{(\mathrm{U})}\left(\NVar{s},\NVar{t};\NVar{\mathbf{x}}\right)$: elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$ ; §36.2(i)
$\Phi^{(1)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x}% ,\NVar{y}\right)$: first $q$ -Appell function; (17.4.5)
$\Phi^{(2)}\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};\NVar{c},\NVar{c^{\prime}}% ;\NVar{q};\NVar{x},\NVar{y}\right)$: second $q$ -Appell function; (17.4.6)
$\Phi^{(3)}\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},\NVar{b^{\prime}};\NVar{c}% ;\NVar{q};\NVar{x},\NVar{y}\right)$: third $q$ -Appell function; (17.4.7)
$\Phi^{(4)}\left(\NVar{a},\NVar{b};\NVar{c},\NVar{c^{\prime}};\NVar{q};\NVar{x}% ,\NVar{y}\right)$: fourth $q$ -Appell function; (17.4.8)
${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left(\NVar{a_{0},\dots,a_{r}};\NVar{b_{1},% \dots,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r+1}}\phi_{\NVar{s}}}\left({\NVar{a_{0},\dots,a_{r}}\atop\NVar{b_{1% },\dots,b_{s}}};\NVar{q},\NVar{z}\right)$: basic hypergeometric (or $q$ -hypergeometric) function; (17.4.1)
$\pi$: the ratio of the circumference of a circle to its diameter; (3.12.1)
$\pi$: set of plane partitions; §26.12(i)
$\pi\left(\NVar{x}\right)$: number of primes not exceeding a number; (27.2.2)
$\Pi(\NVar{z-1})=\Gamma\left(z\right)$: notation used by Gauss; §5.1
(with $\Gamma\left(\NVar{z}\right)$ : gamma function)
$\Pi_{\NVar{m}}(\NVar{a})=\Gamma_{m}\left(a+\tfrac{1}{2}(m+1)\right)$: notation used by Herz (1955, p. 480); §35.1
(with $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function)
$\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s complete elliptic integral of the third kind; (19.2.8)
$\Pi(\NVar{n}\backslash\NVar{\alpha})=\Pi\left(\alpha^{2},k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind)
$\Pi_{1}(\NVar{\nu},\NVar{k})=\Pi\left(\alpha^{2},k\right)$: notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
(with $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind)
$\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$: Legendre’s incomplete elliptic integral of the third kind; (19.2.7)
$\Pi(\NVar{\phi},\NVar{\nu},\NVar{k})=\Pi\left(\phi,\alpha^{2},k\right)$: notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
(with $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind)
$\operatorname{pp}\left(\NVar{n}\right)$: number of plane partitions of $n$ ; §26.12(i)
$\operatorname{pq}\left(\NVar{z},\NVar{k}\right)$: generic Jacobian elliptic function; (22.2.10)
$\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$: spheroidal wave function of the first kind; §30.4(i)
$\mathrm{ps}^{\NVar{m}}_{\NVar{n}}(\NVar{x},\NVar{\gamma^{2}})=\mathsf{Ps}^{m}_% {n}\left(x,\gamma^{2}\right)$: notation used by Meixner and Schäfke (1954) for the spheroidal wave function of the first kind; §30.1
(with $\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{x},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of the first kind)
$\mathrm{Ps}^{\NVar{m}}_{\NVar{n}}(\NVar{z},\NVar{\gamma^{2}})=\mathit{Ps}^{m}_% {n}\left(z,\gamma^{2}\right)$: notation used by Meixner and Schäfke (1954) for the spheroidal wave function of complex argument; §30.1
(with $\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$ : spheroidal wave function of complex argument)
$\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$: spheroidal wave function of complex argument; §30.6
$\psi\left(\NVar{x}\right)$: Chebyshev $\psi$ -function; (25.16.1)
$\Psi(\NVar{z})=\psi\left(z\right)$: notation used by Davis (1933); §5.1
(with $\psi\left(\NVar{z}\right)$ : psi (or digamma) function)
$\Psi(\NVar{z-1})=\psi\left(z\right)$: notation used by Gauss, Jahnke and Emde (1945); §5.1
(with $\psi\left(\NVar{z}\right)$ : psi (or digamma) function)
$\psi\left(\NVar{z}\right)$: psi (or digamma) function; (5.2.2)
$\Psi_{\NVar{K}}\left(\NVar{\mathbf{x}}\right)$: canonical integral function; (36.2.4)
$\Psi_{2}\left(\NVar{\mathbf{x}}\right)$: Pearcey integral; (36.2.14)
$\Psi^{(\mathrm{E})}\left(\NVar{\mathbf{x}}\right)$: elliptic umbilic canonical integral function; (36.2.5)
$\Psi^{(\mathrm{H})}\left(\NVar{\mathbf{x}}\right)$: hyperbolic umbilic canonical integral function; (36.2.5)
$\psi^{(\NVar{n})}\left(\NVar{z}\right)$: polygamma functions; §5.15
$\Psi^{(\mathrm{U})}\left(\NVar{\mathbf{x}}\right)$: umbilic canonical integral function; (36.2.5)
$\Psi_{\NVar{K}}(\NVar{\mathbf{x}};k)$: diffraction catastrophe; (36.2.10)
$\Psi^{(\mathrm{E})}(\NVar{\mathbf{x}};\NVar{k})$: elliptic umbilic canonical integral function; (36.2.11)
$\Psi^{(\mathrm{H})}(\NVar{\mathbf{x}};\NVar{k})$: hyperbolic umbilic canonical integral function; (36.2.11)
$\Psi^{(\mathrm{U})}(\NVar{\mathbf{x}};\NVar{k})$: umbilic canonical integral function; (36.2.11)
$\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$: confluent hypergeometric function of matrix argument (second kind); (35.6.2)
$\Psi(\NVar{a};\NVar{b};\NVar{z})=U\left(a,b,z\right)$: notation used by Erdélyi et al. (1953a, §6.5); §13.1
(with $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function)
${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ or ${{}_{\NVar{r}}\psi_{\NVar{s}}}\left({\NVar{a_{1},\dots,a_{r}}\atop\NVar{b_{1},% \dots,b_{s}}};\NVar{q},\NVar{z}\right)$: bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function; (17.4.3)

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