# Notations P

*ABCDEFGHIJKLMNO♦P♦QRSTUVWXYZ
$\mbox{P}_{\mbox{\scriptsize I}}$, $\mbox{P}_{\mbox{\scriptsize II}}$, $\mbox{P}_{\mbox{\scriptsize III}}$, $\mbox{P}^{\prime}_{\mbox{\scriptsize III}}$, $\mbox{P}_{\mbox{\scriptsize IV}}$, $\mbox{P}_{\mbox{\scriptsize V}}$, $\mbox{P}_{\mbox{\scriptsize VI}}$
Painlevé transcendents; §32.2(i)
$p\left(\NVar{n}\right)$
total number of partitions of $n$; §26.2
$P(\NVar{z})=\tfrac{1}{2}\operatorname{erfc}\left(-z/\sqrt{2}\right)$
alternative notation for the complementary error function; §7.1
$p_{\NVar{k}}\left(\NVar{n}\right)$
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)
$P_{\NVar{\nu}}\left(\NVar{z}\right)=P^{0}_{\nu}\left(z\right)$
Legendre function of the first kind; §14.2(ii)
$P_{\NVar{z}}(\NVar{a})=\gamma\left(a,z\right)$
notation used by Batchelder (1967, p. 63); §8.1
$\mathsf{P}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\left(\NVar{x}\right)$
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)
$\mathrm{P}_{\NVar{\nu}}^{\NVar{\mu}}(\NVar{x})=\mathsf{P}^{\mu}_{\nu}\left(x\right)$
notation used by Erdélyi et al. (1953a), Olver (1997b); §14.1
$\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}\right)$
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
$P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)$
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
$P\left(\NVar{a},\NVar{z}\right)$
normalized incomplete gamma function; 8.2.4
$p\left(\NVar{\mathrm{condition}},\NVar{n}\right)$
restricted number of partions of $n$; §26.10(i)
$\wp\left(\NVar{z}|\NVar{\mathbb{L}}\right)$
Weierstrass $\wp$-function; §23.1
$p_{\NVar{k}}\left(\leq\NVar{m},\NVar{n}\right)$
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); Curtis (1964a):
$P_{\NVar{n}}\left(\NVar{x};\NVar{c}\right)$
associated Legendre polynomial; 18.30.6
$P^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x};\NVar{\phi}\right)$
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
$\wp\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$
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
$p_{\NVar{n}}\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},\NVar{d}\,|\,\NVar{q}\right)$
$P\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$
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
$\phi_{\NVar{k}}\left(\NVar{n}\right)$
sum of powers of integers relatively prime to $n$; 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
$\phi\left(\NVar{z},\NVar{s}\right)=\mathrm{Li}_{s}\left(z\right)$
notation used by (Truesdell, 1945); §25.12(ii)
$\varphi_{\NVar{n},\NVar{m}}\left(\NVar{z},\NVar{q}\right)$
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
$\phi\left(\NVar{\rho},\NVar{\beta};\NVar{z}\right)$
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 $x$; 27.2.2
$\Pi(\NVar{z-1})=\Gamma\left(z\right)$
notation used by Gauss; §5.1
$\Pi_{\NVar{m}}(\NVar{a})=\Gamma_{m}\left(a+\tfrac{1}{2}(m+1)\right)$
notation used by Herz (1955, p. 480); §35.1
$\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$
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
$\Pi_{1}(\NVar{\nu},\NVar{k})=\Pi\left(\alpha^{2},k\right)$
notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
$\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$
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
$\mathrm{pp}\left(\NVar{n}\right)$
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
$\mathrm{Ps}^{\NVar{m}}_{\NVar{n}}(\NVar{z},\NVar{\gamma^{2}})=\mathit{Ps}^{m}_% {n}\left(z,\gamma^{2}\right)$
notation used by Meixner and Schäfke (1954) for the spheroidal wave function of complex argument; §30.1
$\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\left(\NVar{z},\NVar{\gamma^{2}}\right)$
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
$\Psi(\NVar{z-1})=\psi\left(z\right)$
notation used by Gauss, Jahnke and Emde (1945); §5.1
$\psi\left(\NVar{z}\right)$
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
${{}_{\NVar{r}}\psi_{\NVar{s}}}\left(\NVar{a_{1},\dots,a_{r}};\NVar{b_{1},\dots% ,b_{s}};\NVar{q},\NVar{z}\right)$ 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