DLMF: Notations P

Notations P

♦*♦A♦B♦C♦D♦E♦F♦G♦H♦I♦J♦K♦L♦M♦N♦O♦P♦Q♦R♦S♦T♦U♦V♦W♦X♦Y♦Z♦

$\mbox{P}_{{\mbox{\scriptsize I}}}$ , $\mbox{P}_{{\mbox{\scriptsize II}}}$ , $\mbox{P}_{{\mbox{\scriptsize III}}}$ , $\mbox{P}^{{\prime}}_{{\mbox{\scriptsize III}}}$ , $\mbox{P}_{{\mbox{\scriptsize IV}}}$ , $\mbox{P}_{{\mbox{\scriptsize V}}}$ , $\mbox{P}_{{\mbox{\scriptsize VI}}}$: Painlevé transcendents; §32.2(i)
$P(z)=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(-z/\sqrt{2}\right)$: alternative notation for the complementary error function; §7.1
(with $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function)
$\mathop{p\/}\nolimits\!\left(\mathrm{condition},n\right)$: restricted number of partions of $n$ ; §26.10(i)
$\mathop{P\/}\nolimits\!\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}$: Riemann’s $\mathop{P\/}\nolimits$ -symbol for solutions of the generalized hypergeometric differential equation; (15.11.3)
$\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function; (23.2.4)
$\mathop{p\/}\nolimits\!\left(n\right)$: total number of partitions of $n$ ; ¶ ‣ §26.2
$P_{z}(a)=\mathop{\gamma\/}\nolimits\!\left(a,z\right)$: notation used by Batchelder (1967, p. 63); §8.1
(with $\mathop{\gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function)
$\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)$ : $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ with $\mu=0$: ; §14.1
$\mathop{\mathsf{P}_{{\nu}}\/}\nolimits\!\left(x\right)$ : $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ with $\mu=0$: ; ¶ ‣ §14.3(i)
$\mathop{P_{{\nu}}\/}\nolimits\!\left(z\right)$ : $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ with $\mu=0$: ; §14.1
$\mathop{P_{{\nu}}\/}\nolimits\!\left(z\right)$ : $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ with $\mu=0$: ; ¶ ‣ §14.3(ii)
$\mathop{P_{{n}}\/}\nolimits\!\left(x\right)$: Legendre polynomial; Table 18.3.1
$\mathop{P_{{\nu}}\/}\nolimits\!\left(z\right)$ : $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ with $\mu=0$: ; §14.21(i)
$\mathop{p_{{k}}\/}\nolimits\!\left(n\right)$: number of partitions of $n$ into at most $k$ parts; §26.9(i)
$\mathrm{P}_{\nu}^{\mu}(x)=\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$: notation used by Erdélyi et al. (1953a), Olver (1997b); §14.1
(with $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind)
$\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$: Ferrers function of the first kind; (14.3.1)
$\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$: associated Legendre function of the first kind; (14.3.6)
$\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$: associated Legendre function of the first kind; §14.21(i)
$\mathop{P^{{*}}_{{n}}\/}\nolimits\!\left(x\right)$: shifted Legendre polynomial; Table 18.3.1
$P_{{\nu}}^{{\mu}}(x)=\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$: notation used by Magnus et al. (1966); §14.1
(with $\mathop{\mathsf{P}^{{\mu}}_{{\nu}}\/}\nolimits\!\left(x\right)$ : Ferrers function of the first kind)
$\mathfrak{P}_{{\nu}}^{{\mu}}(z)=\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$: notation used by Magnus et al. (1966); §14.1
(with $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind)
$P_{n}^{{(\lambda)}}(x)=\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$: notation used by Szegö (1975, §4.7); §18.1(iii)
(with $\mathop{C^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x\right)$ : ultraspherical (or Gegenbauer) polynomial)
$\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$: Jacobi polynomial; Table 18.3.1
$\mathop{\mathsf{P}^{{-\mu}}_{{-\frac{1}{2}+i\tau}}\/}\nolimits\!\left(x\right)$: conical function; §14.20(i)
$\mathop{P\/}\nolimits\!\left(a,z\right)$: normalized incomplete gamma function; (8.2.4)
$\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$: Weierstrass $\mathop{\wp\/}\nolimits$ -function; §23.1
$P_{\ell}(\epsilon,r)=(2\ell+1)!\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)/2^{{\ell+1}}$: notation used by Curtis (1964a); ¶ ‣ §33.1
(with $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function and $!$ : $n!$ : factorial)
$\Pi _{1}(\nu,k)=\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$: notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
(with $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind)
$\mathop{P_{{n}}\/}\nolimits\!\left(x;c\right)$: associated Legendre polynomial; (18.30.6)
$\mathop{p_{{k}}\/}\nolimits\!\left(\mathcal{D},n\right)$: number of partitions of $n$ into at most $k$ distinct parts; §26.10(i)
$\mathop{p_{{k}}\/}\nolimits\!\left(\leq m,n\right)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ ; §26.9(i)
$\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c\right)$: associated Jacobi polynomial; (18.30.4)
$\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;\phi\right)$: Meixner–Pollaczek polynomial; ¶ ‣ §18.19
$\mathop{P^{{\alpha,\beta,\gamma}}_{{m,n}}\/}\nolimits\!\left(x,y\right)$: triangle polynomial; (18.37.7)
$\Pi(n;\phi\backslash\alpha)=\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind)
$\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$: Weierstrass $\mathop{\wp\/}\nolimits$ -function; (23.3.8)
$\mathop{P^{{(\lambda)}}_{{n}}\/}\nolimits\!\left(x;a,b\right)$: Pollaczek polynomial; (18.35.4)
$\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b;q\right)$: little $q$ -Jacobi polynomial; (18.27.13)
$\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x;c,d;q\right)$: big $q$ -Jacobi polynomial; (18.27.6)
$\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,\conj{a},\conj{b}\right)$: continuous Hahn polynomial; ¶ ‣ §18.19
$\mathop{P_{{n}}\/}\nolimits\!\left(x;a,b,c;q\right)$: big $q$ -Jacobi polynomial; (18.27.5)
$\mathop{p_{{n}}\/}\nolimits\!\left(x;a,b,c,d\,|\, q\right)$: Askey–Wilson polynomial; (18.28.1)
$\mathop{\mathrm{ph}\/}\nolimits$: phase; (1.9.7)
$\mathop{\phi\/}\nolimits\!\left(z\right)$: Airy phase function; §9.8(i)
$\Phi(z)=\tfrac{1}{2}\mathop{\mathrm{erfc}\/}\nolimits\!\left(-z/\sqrt{2}\right)$: alternative notation for the complementary error function; §7.1
(with $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function)
$\mathop{\phi\/}\nolimits\!\left(n\right)$: Euler’s totient; (27.2.7)
$\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$: phase of derivatives of Bessel functions; (10.18.3)
$\mathop{\Phi _{{1}}\/}\nolimits\!\left(t;\mathbf{x}\right)$: fold catastrophe; (36.2.1)
$\mathop{\Phi _{{2}}\/}\nolimits\!\left(t;\mathbf{x}\right)$: cusp catastrophe; (36.2.1)
$\mathop{\Phi _{{3}}\/}\nolimits\!\left(t;\mathbf{x}\right)$: swallowtail catastrophe; (36.2.1)
$\mathop{\Phi _{{K}}\/}\nolimits\!\left(t;\mathbf{x}\right)$: cuspoid catastrophe; (36.2.1)
$\mathop{\phi _{{k}}\/}\nolimits\!\left(n\right)$: sum of powers of integers relatively prime to $n$ ; (27.2.6)
$\mathop{\phi^{{(\alpha,\beta)}}_{{\lambda}}\/}\nolimits\!\left(t\right)$: Jacobi function; (15.9.11)

$\mathop{\phi\/}\nolimits\!\left(z,s\right)=\mathop{\mathrm{Li}_{{s}}\/}\nolimits\!\left(z\right)$: notation used by (Truesdell, 1945); §25.12(ii)
(with $\mathop{\mathrm{Li}_{{s}}\/}\nolimits\!\left(z\right)$ : polylogarithm)
$\mathop{\varphi _{{n,m}}\/}\nolimits\!\left(z,q\right)$: combined theta function; §20.11(v)
$\mathop{\Phi^{{(\mathrm{E})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$: elliptic umbilic catastrophe; (36.2.2)
$\mathop{\Phi^{{(\mathrm{H})}}\/}\nolimits\!\left(s,t;\mathbf{x}\right)$: hyperbolic umbilic catastrophe; (36.2.3)
$\mathop{\phi\/}\nolimits\!\left(\rho,\beta;z\right)$: generalized Bessel function; (10.46.1)
$\Phi(a;b;z)=\mathop{M\/}\nolimits\!\left(a,b,z\right)$: notation used by Humbert (1920); §13.1
(with $\mathop{M\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function)
$\mathop{\Phi\/}\nolimits\!\left(z,s,a\right)$: Lerch’s transcendent; (25.14.1)
$\mathop{\Phi^{{(1)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c;x,y\right)$: first $q$ -Appell function; (17.4.5)
$\mathop{\Phi^{{(2)}}\/}\nolimits\!\left(a;b,b^{{\prime}};c,c^{{\prime}};x,y\right)$: second $q$ -Appell function; (17.4.6)
$\mathop{\Phi^{{(3)}}\/}\nolimits\!\left(a,a^{{\prime}};b,b^{{\prime}};c;x,y\right)$: third $q$ -Appell function; (17.4.7)
$\mathop{\Phi^{{(4)}}\/}\nolimits\!\left(a;b;c,c^{{\prime}};x,y\right)$: fourth $q$ -Appell function; (17.4.8)
$\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left({a_{0},a_{1},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$: basic hypergeometric (or $q$ -hypergeometric) function; (17.4.1)
$\mathop{{{}_{{r+1}}\phi _{{s}}}\/}\nolimits\!\left(a_{0},a_{1},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$: basic hypergeometric (or $q$ -hypergeometric) function; (17.4.1)
$\pi$: set of plane partitions; §26.12(i)
$\Pi(z-1)=\mathop{\Gamma\/}\nolimits\!\left(z\right)$: notation used by Gauss; §5.1
(with $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function)
$\mathop{\pi\/}\nolimits\!\left(x\right)$: number of primes not exceeding $x$ ; (27.2.2)
$\Pi _{m}(a)=\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a+\tfrac{1}{2}(m+1)\right)$: notation used by Herz (1955, p. 480); §35.1
(with $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function)
$\Pi(n\backslash\alpha)=\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$: notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind)
$\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$: Legendre’s complete elliptic integral of the third kind; (19.2.8)
$\Pi(\phi,\nu,k)=\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$: notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
(with $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind)
$\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$: Legendre’s incomplete elliptic integral of the third kind; (19.2.7)
$\mathop{\mathit{pp}\/}\nolimits\!\left(n\right)$: number of plane partitions of $n$ ; §26.12(i)
$\mathop{\mathrm{pq}\/}\nolimits\left(z,k\right)$: generic Jacobian elliptic function; (22.2.10)
$\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$: spheroidal wave function of complex argument; §30.6
$\mathrm{Ps}^{{m}}_{{n}}(z,\gamma^{2})=\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$: notation used by Meixner and Schäfke (1954) for the spheroidal wave function of complex argument; §30.1
(with $\mathop{\mathit{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(z,\gamma^{2}\right)$ : spheroidal wave function of complex argument)
$\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$: spheroidal wave function of the first kind; §30.4(i)
$\mathrm{ps}^{{m}}_{{n}}(x,\gamma^{2})=\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$: notation used by Meixner and Schäfke (1954) for the spheroidal wave function of the first kind; §30.1
(with $\mathop{\mathsf{Ps}^{{m}}_{{n}}\/}\nolimits\!\left(x,\gamma^{2}\right)$ : spheroidal wave function of the first kind)
$\mathop{\psi\/}\nolimits\!\left(x\right)$: Chebyshev $\mathop{\psi\/}\nolimits$ -function; (25.16.1)
$\Psi(z)=\mathop{\psi\/}\nolimits\!\left(z\right)$: notation used by Davis (1933); §5.1
(with $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function)
$\Psi(z-1)=\mathop{\psi\/}\nolimits\!\left(z\right)$: notation used by Gauss, Jahnke and Emde (1945); §5.1
(with $\mathop{\psi\/}\nolimits\!\left(z\right)$ : psi (or digamma) function)
$\mathop{\psi\/}\nolimits\!\left(z\right)$: psi (or digamma) function; (5.2.2)
$\mathop{\Psi _{{2}}\/}\nolimits\!\left(\mathbf{x}\right)$: Pearcey integral; (36.2.14)
$\mathop{\Psi _{{K}}\/}\nolimits\!\left(\mathbf{x}\right)$: canonical integral; (36.2.4)
$\mathop{\psi^{{(n)}}\/}\nolimits\!\left(z\right)$: polygamma functions; §5.15
$\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!\left(\mathbf{x}\right)$: canonical integral; (36.2.6)
$\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!\left(\mathbf{x}\right)$: canonical integral; (36.2.8)
$\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!\left(\mathbf{x}\right)$: canonical umbilic integral; (36.2.5)
$\mathop{\Psi _{{3}}\/}\nolimits\!(\mathbf{x};k)$: swallowtail canonical integral function; (36.2.10)
$\mathop{\Psi _{{3}}\/}\nolimits\!(\mathbf{x};k)$: swallowtail canonical integral function; §36.3(i)
$\mathop{\Psi _{{K}}\/}\nolimits\!(\mathbf{x};k)$: diffraction catastrophe; (36.2.10)
$\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!(\mathbf{x};k)$: elliptic umbilic canonical integral function; §36.3(ii)
$\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!(\mathbf{x};k)$: elliptic umbilic canonical integral function; (36.2.11)
$\mathop{\Psi^{{(\mathrm{E})}}\/}\nolimits\!(\mathbf{x};k)$: elliptic umbilic canonical integral function; §36.3(i)
$\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!(\mathbf{x};k)$: hyperbolic umbilic canonical integral function; §36.3(i)
$\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!(\mathbf{x};k)$: hyperbolic umbilic canonical integral function; §36.3(ii)
$\mathop{\Psi^{{(\mathrm{H})}}\/}\nolimits\!(\mathbf{x};k)$: hyperbolic umbilic canonical integral function; (36.2.11)
$\mathop{\Psi^{{(\mathrm{U})}}\/}\nolimits\!(\mathbf{x};k)$: umbilic canonical integral function; (36.2.11)
$\Psi(a;b;z)=\mathop{U\/}\nolimits\!\left(a,b,z\right)$: notation used by Erdélyi et al. (1953a, §6.5); §13.1
(with $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function)
$\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)$: confluent hypergeometric function of matrix argument (second kind); (35.6.2)
$\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)$: confluent hypergeometric function of matrix argument (second kind); §35.1
$\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits\!\left({a_{1},a_{2},\dots,a_{r}\atop b_{1},b_{2},\dots,b_{s}};q,z\right)$: bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function; (17.4.3)
$\mathop{{{}_{{r}}\psi _{{s}}}\/}\nolimits\!\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$: bilateral basic hypergeometric (or bilateral $q$ -hypergeometric) function; (17.4.3)