# 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)
$\mathop{p\/}\nolimits\!\left(\NVar{n}\right)$
total number of partitions of $n$; §26.2
$P(\NVar{z})=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(-z/\sqrt{2}\right)$
alternative notation for the complementary error function; §7.1
$\mathop{p_{\NVar{k}}\/}\nolimits\!\left(\NVar{n}\right)$
number of partitions of $n$ into at most $k$ parts; §26.9(i)
$\mathop{P_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$
Legendre polynomial; Table 18.3.1
$\mathop{P^{*}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$
shifted Legendre polynomial; Table 18.3.1
$\mathop{\mathsf{P}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)=\mathop{% \mathsf{P}^{0}_{\nu}\/}\nolimits\!\left(x\right)$
Ferrers function of the first kind; §14.2(ii)
$\mathop{P_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)=\mathop{P^{0}_{\nu}% \/}\nolimits\!\left(z\right)$
Legendre function of the first kind; §14.2(ii)
$P_{\NVar{z}}(\NVar{a})=\mathop{\gamma\/}\nolimits\!\left(a,z\right)$
notation used by Batchelder (1967, p. 63); §8.1
$\mathop{\mathsf{P}^{-\NVar{\mu}}_{\NVar{-\frac{1}{2}+i\tau}}\/}\nolimits\!% \left(\NVar{x}\right)$
conical function; §14.20(i)
$\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% }\right)$
Jacobi polynomial; Table 18.3.1
$P_{\NVar{n}}^{(\NVar{\lambda})}(\NVar{x})=\mathop{C^{(\lambda)}_{n}\/}% \nolimits\!\left(x\right)$
notation used by Szegő (1975, §4.7); §18.1(iii)
$\mathrm{P}_{\NVar{\nu}}^{\NVar{\mu}}(\NVar{x})=\mathop{\mathsf{P}^{\mu}_{\nu}% \/}\nolimits\!\left(x\right)$
notation used by Erdélyi et al. (1953a), Olver (1997b); §14.1
$\mathop{\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$
Ferrers function of the first kind; 14.3.1
$P_{\NVar{\nu}}^{\NVar{\mu}}(\NVar{x})=\mathop{\mathsf{P}^{\mu}_{\nu}\/}% \nolimits\!\left(x\right)$
notation used by Magnus et al. (1966); §14.1
$\mathop{P^{\NVar{\mu}}_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$
associated Legendre function of the first kind; §14.21(i)
$\mathfrak{P}_{\NVar{\nu}}^{\NVar{\mu}}(\NVar{z})=\mathop{P^{\mu}_{\nu}\/}% \nolimits\!\left(z\right)$
notation used by Magnus et al. (1966); §14.1
$\mathop{P\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$
normalized incomplete gamma function; 8.2.4
$\mathop{p\/}\nolimits\!\left(\NVar{\mathrm{condition}},\NVar{n}\right)$
restricted number of partions of $n$; §26.10(i)
$\mathop{\wp\/}\nolimits\!\left(\NVar{z}|\NVar{\mathbb{L}}\right)$
Weierstrass $\mathop{\wp\/}\nolimits$-function; §23.1
$\mathop{p_{\NVar{k}}\/}\nolimits\!\left(\leq\NVar{m},\NVar{n}\right)$
number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$; §26.9(i)
$\mathop{p_{\NVar{k}}\/}\nolimits\!\left(\NVar{\mathcal{D}},\NVar{n}\right)$
number of partitions of $n$ into at most $k$ distinct parts; §26.10(i)
$P_{\NVar{\ell}}(\NVar{\epsilon},\NVar{r})=(2\ell+1)!\mathop{f\/}\nolimits\!% \left(\epsilon,\ell;r\right)/2^{\ell+1}$
notation used by Curtis (1964a); Curtis (1964a):
$\mathop{P_{\NVar{n}}\/}\nolimits\!\left(\NVar{x};\NVar{c}\right)$
associated Legendre polynomial; 18.30.6
$\mathop{P^{(\NVar{\lambda})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x};\NVar{\phi% }\right)$
Meixner–Pollaczek polynomial; §18.19
$\mathop{P^{\NVar{\alpha},\NVar{\beta},\NVar{\gamma}}_{\NVar{m},\NVar{n}}\/}% \nolimits\!\left(\NVar{x},\NVar{y}\right)$
triangle polynomial; 18.37.7
$\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% };\NVar{c}\right)$
associated Jacobi polynomial; 18.30.4
$\Pi(\NVar{n};\NVar{\phi}\backslash\NVar{\alpha})=\mathop{\Pi\/}\nolimits\!% \left(\phi,\alpha^{2},k\right)$
notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
$\mathop{\wp\/}\nolimits\!\left(\NVar{z};\NVar{g_{2}},\NVar{g_{3}}\right)$
Weierstrass $\mathop{\wp\/}\nolimits$-function; 23.3.8
$\mathop{P^{(\NVar{\lambda})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x};\NVar{a},% \NVar{b}\right)$
Pollaczek polynomial; 18.35.4
$\mathop{p_{\NVar{n}}\/}\nolimits\!\left(\NVar{x};\NVar{a},\NVar{b};\NVar{q}\right)$
little $q$-Jacobi polynomial; 18.27.13
$\mathop{P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x% };\NVar{c},\NVar{d};\NVar{q}\right)$
big $q$-Jacobi polynomial; 18.27.6
$\mathop{p_{\NVar{n}}\/}\nolimits\!\left(\NVar{x};\NVar{a},\NVar{b},\NVar{% \overline{a}},\NVar{\overline{b}}\right)$
continuous Hahn polynomial; §18.19
$\mathop{P_{\NVar{n}}\/}\nolimits\!\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c};% \NVar{q}\right)$
big $q$-Jacobi polynomial; 18.27.5
$\mathop{p_{\NVar{n}}\/}\nolimits\!\left(\NVar{x};\NVar{a},\NVar{b},\NVar{c},% \NVar{d}\,|\,\NVar{q}\right)$
$\mathop{P\/}\nolimits\!\NVar{\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}}$
Riemann’s $\mathop{P\/}\nolimits$-symbol for solutions of the generalized hypergeometric differential equation; 15.11.3
$\mathop{\mathrm{ph}\/}\nolimits$
phase; 1.9.7
$\mathop{\phi\/}\nolimits\!\left(\NVar{n}\right)$
Euler’s totient; 27.2.7
$\mathop{\phi\/}\nolimits\!\left(\NVar{z}\right)$
Airy phase function; §9.8(i)
$\Phi(\NVar{z})=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(-z/\sqrt{2% }\right)$
alternative notation for the complementary error function; §7.1
$\mathop{\phi_{\NVar{k}}\/}\nolimits\!\left(\NVar{n}\right)$
sum of powers of integers relatively prime to $n$; 27.2.6
$\mathop{\phi_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{x}\right)$
phase of derivatives of Bessel functions; 10.18.3
$\mathop{\phi^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{\lambda}}\/}\nolimits\!% \left(\NVar{t}\right)$
Jacobi function; 15.9.11
$\mathop{\phi\/}\nolimits\!\left(\NVar{z},\NVar{s}\right)=\mathop{\mathrm{Li}_{% s}\/}\nolimits\!\left(z\right)$
notation used by (Truesdell, 1945); §25.12(ii)
$\mathop{\varphi_{\NVar{n},\NVar{m}}\/}\nolimits\!\left(\NVar{z},\NVar{q}\right)$
combined theta function; §20.11(v)
$\mathop{\Phi_{\NVar{K}}\/}\nolimits\!\left(\NVar{t};\NVar{\mathbf{x}}\right)$
cuspoid catastrophe of codimension $K$; 36.2.1
$\Phi(\NVar{a};\NVar{b};\NVar{z})=\mathop{M\/}\nolimits\!\left(a,b,z\right)$
notation used by Humbert (1920); §13.1
$\mathop{\phi\/}\nolimits\!\left(\NVar{\rho},\NVar{\beta};\NVar{z}\right)$
generalized Bessel function; 10.46.1
$\mathop{\Phi\/}\nolimits\!\left(\NVar{z},\NVar{s},\NVar{a}\right)$
Lerch’s transcendent; 25.14.1
$\mathop{\Phi^{(\mathrm{E})}\/}\nolimits\!\left(\NVar{s},\NVar{t};\NVar{\mathbf% {x}}\right)$
elliptic umbilic catastrophe; 36.2.2
$\mathop{\Phi^{(\mathrm{H})}\/}\nolimits\!\left(\NVar{s},\NVar{t};\NVar{\mathbf% {x}}\right)$
hyperbolic umbilic catastrophe; 36.2.3
$\mathop{\Phi^{(\mathrm{U})}\/}\nolimits\!\left(\NVar{s},\NVar{t};\NVar{\mathbf% {x}}\right)$
elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\mbox{ or }\mathrm{K}$; §36.2(i)
$\mathop{\Phi^{(1)}\/}\nolimits\!\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};% \NVar{c};\NVar{q};\NVar{x},\NVar{y}\right)$
first $q$-Appell function; 17.4.5
$\mathop{\Phi^{(2)}\/}\nolimits\!\left(\NVar{a};\NVar{b},\NVar{b^{\prime}};% \NVar{c},\NVar{c^{\prime}};\NVar{q};\NVar{x},\NVar{y}\right)$
second $q$-Appell function; 17.4.6
$\mathop{\Phi^{(3)}\/}\nolimits\!\left(\NVar{a},\NVar{a^{\prime}};\NVar{b},% \NVar{b^{\prime}};\NVar{c};\NVar{q};\NVar{x},\NVar{y}\right)$
third $q$-Appell function; 17.4.7
$\mathop{\Phi^{(4)}\/}\nolimits\!\left(\NVar{a},\NVar{b};\NVar{c},\NVar{c^{% \prime}};\NVar{q};\NVar{x},\NVar{y}\right)$
fourth $q$-Appell function; 17.4.8
$\mathop{{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\/}\nolimits\!\left(\NVar{a_{0},\dots,% a_{r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)$ or $\mathop{{{}_{\NVar{r+1}}\phi_{\NVar{s}}}\/}\nolimits\!\left({\NVar{a_{0},\dots% ,a_{r}}\atop\NVar{b_{1},\dots,b_{s}}};\NVar{q},\NVar{z}\right)$
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)
$\mathop{\pi\/}\nolimits\!\left(\NVar{x}\right)$
number of primes not exceeding $x$; 27.2.2
$\Pi(\NVar{z-1})=\mathop{\Gamma\/}\nolimits\!\left(z\right)$
notation used by Gauss; §5.1
$\Pi_{\NVar{m}}(\NVar{a})=\mathop{\Gamma_{m}\/}\nolimits\!\left(a+\tfrac{1}{2}(% m+1)\right)$
notation used by Herz (1955, p. 480); §35.1
$\mathop{\Pi\/}\nolimits\!\left(\NVar{\alpha}^{2},\NVar{k}\right)$
Legendre’s complete elliptic integral of the third kind; 19.2.8
$\Pi(\NVar{n}\backslash\NVar{\alpha})=\mathop{\Pi\/}\nolimits\!\left(\alpha^{2}% ,k\right)$
notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
$\Pi_{1}(\NVar{\nu},\NVar{k})=\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$
notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
$\mathop{\Pi\/}\nolimits\!\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$
Legendre’s incomplete elliptic integral of the third kind; 19.2.7
$\Pi(\NVar{\phi},\NVar{\nu},\NVar{k})=\mathop{\Pi\/}\nolimits\!\left(\phi,% \alpha^{2},k\right)$
notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
$\mathop{\mathrm{pp}\/}\nolimits\!\left(\NVar{n}\right)$
number of plane partitions of $n$; §26.12(i)
$\mathop{\mathrm{p\!q}\/}\nolimits\left(\NVar{z},\NVar{k}\right)$
generic Jacobian elliptic function; 22.2.10
$\mathop{\mathsf{Ps}^{\NVar{m}}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x},\NVar{% \gamma^{2}}\right)$
spheroidal wave function of the first kind; §30.4(i)
$\mathrm{ps}^{\NVar{m}}_{\NVar{n}}(\NVar{x},\NVar{\gamma^{2}})=\mathop{\mathsf{% Ps}^{m}_{n}\/}\nolimits\!\left(x,\gamma^{2}\right)$
notation used by Meixner and Schäfke (1954) for the spheroidal wave function of the first kind; §30.1
$\mathrm{Ps}^{\NVar{m}}_{\NVar{n}}(\NVar{z},\NVar{\gamma^{2}})=\mathop{\mathit{% Ps}^{m}_{n}\/}\nolimits\!\left(z,\gamma^{2}\right)$
notation used by Meixner and Schäfke (1954) for the spheroidal wave function of complex argument; §30.1
$\mathop{\mathit{Ps}^{\NVar{m}}_{\NVar{n}}\/}\nolimits\!\left(\NVar{z},\NVar{% \gamma^{2}}\right)$
spheroidal wave function of complex argument; §30.6
$\mathop{\psi\/}\nolimits\!\left(\NVar{x}\right)$
Chebyshev $\mathop{\psi\/}\nolimits$-function; 25.16.1
$\Psi(\NVar{z})=\mathop{\psi\/}\nolimits\!\left(z\right)$
notation used by Davis (1933); §5.1
$\Psi(\NVar{z-1})=\mathop{\psi\/}\nolimits\!\left(z\right)$
notation used by Gauss, Jahnke and Emde (1945); §5.1
$\mathop{\psi\/}\nolimits\!\left(\NVar{z}\right)$
psi (or digamma) function; 5.2.2
$\mathop{\Psi_{\NVar{K}}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$
canonical integral function; 36.2.4
$\mathop{\Psi_{2}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$
Pearcey integral; 36.2.14
$\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$
elliptic umbilic canonical integral function; 36.2.5
$\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$
hyperbolic umbilic canonical integral function; 36.2.5
$\mathop{\psi^{(\NVar{n})}\/}\nolimits\!\left(\NVar{z}\right)$
polygamma functions; §5.15
$\mathop{\Psi^{(\mathrm{U})}\/}\nolimits\!\left(\NVar{\mathbf{x}}\right)$
umbilic canonical integral function; 36.2.5
$\mathop{\Psi_{\NVar{K}}\/}\nolimits\!(\NVar{\mathbf{x}};k)$
diffraction catastrophe; 36.2.10
$\mathop{\Psi^{(\mathrm{E})}\/}\nolimits\!(\NVar{\mathbf{x}};\NVar{k})$
elliptic umbilic canonical integral function; 36.2.11
$\mathop{\Psi^{(\mathrm{H})}\/}\nolimits\!(\NVar{\mathbf{x}};\NVar{k})$
hyperbolic umbilic canonical integral function; 36.2.11
$\mathop{\Psi^{(\mathrm{U})}\/}\nolimits\!(\NVar{\mathbf{x}};\NVar{k})$
umbilic canonical integral function; 36.2.11
$\mathop{\Psi\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$
confluent hypergeometric function of matrix argument (second kind); 35.6.2
$\Psi(\NVar{a};\NVar{b};\NVar{z})=\mathop{U\/}\nolimits\!\left(a,b,z\right)$
notation used by Erdélyi et al. (1953a, §6.5); §13.1
$\mathop{{{}_{\NVar{r}}\psi_{\NVar{s}}}\/}\nolimits\!\left(\NVar{a_{1},\dots,a_% {r}};\NVar{b_{1},\dots,b_{s}};\NVar{q},\NVar{z}\right)$ or $\mathop{{{}_{\NVar{r}}\psi_{\NVar{s}}}\/}\nolimits\!\left({\NVar{a_{1},\dots,a% _{r}}\atop\NVar{b_{1},\dots,b_{s}}};\NVar{q},\NVar{z}\right)$
bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function; 17.4.3