Notations Notations O Notations Q

Notations P

${\text{P}}_{\text{I}}$, ${\text{P}}_{\text{II}}$, ${\text{P}}_{\text{III}}$, ${\text{P}}_{\text{III}}^{\prime }$, ${\text{P}}_{\text{IV}}$, ${\text{P}}_{\text{V}}$, ${\text{P}}_{\text{VI}}$

Painlevé transcendents; §32.2(i)

$p_k\left(n\right)$

total number of partitions of $n$ into at most $k$ parts; §26.9(i)

$P(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$

alternative notation for the complementary error function; §7.1
(with $\mathrm{erfc}z$: complementary error function)

$p_k\left(n\right)$

number of partitions of $n$ into at most $k$ parts; §26.9(i)

$P_n\left(x\right)$

Legendre polynomial; Table 18.3.1

$P_n^{\ast }\left(x\right)$

shifted Legendre polynomial; Table 18.3.1

${𝖯}_{\nu }\left(x\right)={𝖯}_{\nu }^0\left(x\right)$

Ferrers function of the first kind; §14.2(ii)
(with ${𝖯}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the first kind)

$P_{\nu }\left(z\right)=P_{\nu }^0\left(z\right)$

Legendre function of the first kind; §14.2(ii)
(with $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind)

$P_z(a)=\gamma (a,z)$

notation used by Batchelder (1967, p. 63); §8.1
(with $\gamma (a,z)$: incomplete gamma function)

${𝖯}_{-\frac{1}{2}+\mathrm{i}\tau }^{-\mu }\left(x\right)$

conical function; §14.20(i)

$P_n^{(\alpha ,\beta )}\left(x\right)$

Jacobi polynomial; Table 18.3.1

$P_n^{(\lambda )}(x)=C_n^{(\lambda )}\left(x\right)$

notation used by Szegő (1975, §4.7); §18.1(iii)
(with $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial)

$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$

Jacobi function of matrix argument; (35.7.2)

${\mathrm{P}}_{\nu }^{\mu }(x)={𝖯}_{\nu }^{\mu }\left(x\right)$

notation used by Erdélyi et al. (1953a), Olver (1997b); §14.1
(with ${𝖯}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the first kind)

${𝖯}_{\nu }^{\mu }\left(x\right)$

Ferrers function of the first kind; (14.3.1)

$P_{\nu }^{\mu }(x)={𝖯}_{\nu }^{\mu }\left(x\right)$

notation used by Magnus et al. (1966); §14.1
(with ${𝖯}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the first kind)

$P_{\nu }^{\mu }\left(z\right)$

associated Legendre function of the first kind; §14.21(i)

${𝔓}_{\nu }^{\mu }(z)=P_{\nu }^{\mu }\left(z\right)$

notation used by Magnus et al. (1966); §14.1
(with $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind)

$P(a,z)$

normalized incomplete gamma function; (8.2.4)

$p(\mathrm{condition},n)$

restricted number of partitions of $n$; §26.10(i)

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$)

Weierstrass $\mathrm{\wp }$-function; (23.2.4)

$p_k(\le m,n)$

number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$; §26.9(i)

$p_k(𝒟,n)$

number of partitions of $n$ into at most $k$ distinct parts; §26.10(i)

$P_{\mathrm{\ell }}(ϵ,r)=(2\mathrm{\ell }+1)!f(ϵ,\mathrm{\ell };r)/2^{\mathrm{\ell }+1}$

notation used by Curtis (1964a); item Curtis (1964a):
(with $f(ϵ,\mathrm{\ell };r)$: regular Coulomb function and $!$: factorial (as in $n!$))

$H_n(x;c)$

associated Hermite polynomial; §18.30(iv)

$P_n(x;c)$

associated Legendre polynomial; (18.30.6)

${𝒫}_n^{\lambda }(x;\varphi ,c)$

associated Meixner–Pollaczek polynomial; §18.30(v)

$P_n^{(\lambda )}(x;\varphi )$

Meixner–Pollaczek polynomial; §18.19

$P_{m,n}^{\alpha ,\beta ,\gamma }(x,y)$

triangle polynomial; (18.37.7)

$P_n^{(\alpha ,\beta )}(x;c)$

associated Jacobi polynomial; (18.30.4)

$\mathrm{\Pi }(n;\varphi \backslash \alpha )=\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind)

$\mathrm{\wp }(z;g_2,g_3)$

Weierstrass $\mathrm{\wp }$-function; (23.3.8)

$P_n^{(\lambda )}(x;a,b)$

Pollaczek polynomial; (18.35.4)

$p_n(x;a,b;q)$

little $q$-Jacobi polynomial; (18.27.13)

$P_n^{(\alpha ,\beta )}(x;c,d;q)$

big $q$-Jacobi polynomial; (18.27.6)

$p_n(x;a,b,\overline{a},\overline{b})$

continuous Hahn polynomial; §18.19

$P_n(x;a,b,c;q)$

big $q$-Jacobi polynomial; (18.27.5)

$p_n(x;a,b,c,d|q)$

Askey–Wilson polynomial; (18.28.1)

Riemann’s $P$-symbol for solutions of the generalized hypergeometric differential equation; (15.11.3)

$\mathrm{ph}$

phase; (1.9.7)

$\varphi \left(n\right)$

Euler’s totient; (27.2.7)

$\varphi \left(z\right)$

Airy phase function; (9.8.8)

$\mathrm{\Phi }(z)=\frac{1}{2}\mathrm{erfc}\left(-z/\sqrt{2}\right)$

alternative notation for the complementary error function; §7.1
(with $\mathrm{erfc}z$: complementary error function)

${\varphi }_k\left(n\right)$

sum of powers of integers relatively prime to a number; (27.2.6)

${\varphi }_{\nu }\left(x\right)$

phase of derivatives of Bessel functions; (10.18.3)

${\varphi }_{\lambda }^{(\alpha ,\beta )}\left(t\right)$

Jacobi function; (15.9.11)

$\varphi (z,s)={\mathrm{Li}}_s\left(z\right)$

notation used by (Truesdell, 1945); §25.12(ii)
(with ${\mathrm{Li}}_s\left(z\right)$: polylogarithm)

${\phi }_{n,m}(z,q)$

combined theta function; §20.11(v)

${\mathrm{\Phi }}_K(t;𝐱)$

cuspoid catastrophe of codimension $K$; (36.2.1)

$\mathrm{\Phi }(a;b;z)=M(a,b,z)$

notation used by Humbert (1920); §13.1
(with $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function)

$\varphi (\rho ,\beta ;z)$

generalized Bessel function; (10.46.1)

$\mathrm{\Phi }(z,s,a)$

Lerch’s transcendent; (25.14.1)

${\mathrm{\Phi }}^{(\mathrm{E})}(s,t;𝐱)$

elliptic umbilic catastrophe; (36.2.2)

${\mathrm{\Phi }}^{(\mathrm{H})}(s,t;𝐱)$

hyperbolic umbilic catastrophe; (36.2.3)

elliptic umbilic catastrophe for $\mathrm{U}=\mathrm{E}\text{ or }\mathrm{K}$; §36.2(i)

${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$

first $q$-Appell function; (17.4.5)

${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$

second $q$-Appell function; (17.4.6)

${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$

third $q$-Appell function; (17.4.7)

${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$

fourth $q$-Appell function; (17.4.8)

${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$

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)

$\pi \left(x\right)$

number of primes not exceeding a number; (27.2.2)

$\mathrm{\Pi }(z-1)=\mathrm{\Gamma }\left(z\right)$

notation used by Gauss; §5.1
(with $\mathrm{\Gamma }\left(z\right)$: gamma function)

${\mathrm{\Pi }}_m(a)={\mathrm{\Gamma }}_m\left(a+\frac{1}{2}(m+1)\right)$

notation used by Herz (1955, p. 480); §35.1
(with ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function)

$\mathrm{\Pi }({\alpha }^2,k)$

Legendre’s complete elliptic integral of the third kind; (19.2.8)

$\mathrm{\Pi }(n\backslash \alpha )=\mathrm{\Pi }({\alpha }^2,k)$

notation used by Abramowitz and Stegun (1964, Chapter 17); §19.1
(with $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind)

${\mathrm{\Pi }}_1(\nu ,k)=\mathrm{\Pi }({\alpha }^2,k)$

notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
(with $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind)

$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

Legendre’s incomplete elliptic integral of the third kind; (19.2.7)

$\mathrm{\Pi }(\varphi ,\nu ,k)=\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

notation used by Erdélyi et al. (1953b, Chapter 13); §19.1
(with $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind)

$\mathrm{pp}\left(n\right)$

number of plane partitions of $n$; §26.12(i)

$\mathrm{p}\mathrm{q}(z,k)$

generic Jacobian elliptic function; (22.2.10)

${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$

spheroidal wave function of the first kind; §30.4(i)

${\mathrm{ps}}_n^m(x,{\gamma }^2)={\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$

notation used by Meixner and Schäfke (1954) for the spheroidal wave function of the first kind; §30.1
(with ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind)

${\mathrm{Ps}}_n^m(z,{\gamma }^2)={\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$

notation used by Meixner and Schäfke (1954) for the spheroidal wave function of complex argument; §30.1
(with ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument)

${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$

spheroidal wave function of complex argument; §30.6

$\psi \left(x\right)$

Chebyshev $\psi$-function; (25.16.1)

$\mathrm{\Psi }(z)=\psi \left(z\right)$

notation used by Davis (1933); §5.1
(with $\psi \left(z\right)$: psi (or digamma) function)

$\mathrm{\Psi }(z-1)=\psi \left(z\right)$

notation used by Gauss, Jahnke and Emde (1945); §5.1
(with $\psi \left(z\right)$: psi (or digamma) function)

$\psi \left(z\right)$

psi (or digamma) function; (5.2.2)

${\mathrm{\Psi }}_K\left(𝐱\right)$

canonical integral function; (36.2.4)

${\mathrm{\Psi }}_2\left(𝐱\right)$

Pearcey integral; (36.2.14)

${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$

elliptic umbilic canonical integral function; (36.2.5)

${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$

hyperbolic umbilic canonical integral function; (36.2.5)

${\psi }^{(n)}\left(z\right)$

polygamma functions; §5.15

${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$

umbilic canonical integral function; (36.2.5)

${\mathrm{\Psi }}_K(𝐱;k)$

diffraction catastrophe; (36.2.10)

${\mathrm{\Psi }}^{(\mathrm{E})}(𝐱;k)$

elliptic umbilic canonical integral function; (36.2.11)

${\mathrm{\Psi }}^{(\mathrm{H})}(𝐱;k)$

hyperbolic umbilic canonical integral function; (36.2.11)

${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$

umbilic canonical integral function; (36.2.11)

$\mathrm{\Psi }(a;b;𝐓)$

confluent hypergeometric function of matrix argument (second kind); (35.6.2)

$\mathrm{\Psi }(a;b;z)=U(a,b,z)$

notation used by Erdélyi et al. (1953a, §6.5); §13.1
(with $U(a,b,z)$: Kummer confluent hypergeometric function)

${}_r{}^{}\psi _s^{}(a_1,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_r{}^{}\psi _s^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$

bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function; (17.4.3)

Notations O Notations Q

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