Digital Library of Mathematical Functions
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Notations

Notations P

*ABCDEFGHIJKLMNO♦P♦QRSTUVWXYZ
PI, PII, PIII, PIII, PIV, PV, PVI
Painlevé transcendents; 32.2(i)
P(z)=12erfc(-z/2)
alternative notation for the complementary error function; 7.1
(with erfcz: complementary error function)
p(condition,n)
restricted number of partions of n; 26.10(i)
P{αβγa1b1c1za2b2c2}
Riemann’s P-symbol for solutions of the generalized hypergeometric differential equation; (15.11.3)
(z) (= (z|𝕃) = (z;g2,g3))
Weierstrass -function; (23.2.4)
p(n)
total number of partitions of n; 26.2
Pz(a)=γ(a,z)
notation used by Batchelder (1967, p. 63); 8.1
(with γ(a,z): incomplete gamma function)
Pν(x): Pνμ(x) with μ=0
; 14.1
Pν(x): Pνμ(x) with μ=0
; 14.3(i)
Pν(z): Pνμ(z) with μ=0
; 14.1
Pν(z): Pνμ(z) with μ=0
; 14.21(i)
Pν(z): Pνμ(z) with μ=0
; 14.3(ii)
Pn(x)
Legendre polynomial; 18.3.1
pk(n)
number of partitions of n into at most k parts; 26.9(i)
Pνμ(x)=Pνμ(x)
notation used by Erdélyi et al. (1953a), Olver (1997b); 14.1
(with Pνμ(x): Ferrers function of the first kind)
Pνμ(x)
Ferrers function of the first kind; (14.3.1)
Pνμ(z)
associated Legendre function of the first kind; 14.21(i)
Pνμ(z)
associated Legendre function of the first kind; (14.3.6)
Pn*(x)
shifted Legendre polynomial; 18.3.1
Pνμ(x)=Pνμ(x)
notation used by Magnus et al. (1966); 14.1
(with Pνμ(x): Ferrers function of the first kind)
𝔓νμ(z)=Pνμ(z)
notation used by Magnus et al. (1966); 14.1
(with Pνμ(z): associated Legendre function of the first kind)
Pn(λ)(x)=Cn(λ)(x)
notation used by Szegö (1975, §4.7); 18.1(iii)
(with Cn(λ)(x): ultraspherical (or Gegenbauer) polynomial)
Pn(α,β)(x)
Jacobi polynomial; 18.3.1
P-12+τ-μ(x)
conical function; 14.20(i)
P(a,z)
normalized incomplete gamma function; (8.2.4)
(z|𝕃)
Weierstrass -function; 23.1
P(ϵ,r)=(2+1)!f(ϵ,;r)/2+1
notation used by Curtis (1964a); Curtis (1964a):
(with f(ϵ,;r): regular Coulomb function and !: n!: factorial)
Π1(ν,k)=Π(α2,k)
notation used by Erdélyi et al. (1953b, Chapter 13); 19.1
(with Π(α2,k): Legendre’s complete elliptic integral of the third kind)
Pn(x;c)
associated Legendre polynomial; (18.30.6)
pk(𝒟,n)
number of partitions of n into at most k distinct parts; 26.10(i)
pk(m,n)
number of partitions of n into at most k parts, each less than or equal to m; 26.9(i)
Pn(α,β)(x;c)
associated Jacobi polynomial; (18.30.4)
Pn(λ)(x;ϕ)
Meixner–Pollaczek polynomial; 18.19
Pm,nα,β,γ(x,y)
triangle polynomial; (18.37.7)
Π(n;ϕ\α)=Π(ϕ,α2,k)
notation used by Abramowitz and Stegun (1964, Chapter 17); 19.1
(with Π(ϕ,α2,k): Legendre’s incomplete elliptic integral of the third kind)
(z;g2,g3)
Weierstrass -function; (23.3.8)
Pn(λ)(x;a,b)
Pollaczek polynomial; (18.35.4)
pn(x;a,b;q)
little q-Jacobi polynomial; (18.27.13)
Pn(α,β)(x;c,d;q)
big q-Jacobi polynomial; (18.27.6)
pn(x;a,b,a¯,b¯)
continuous Hahn polynomial; 18.19
Pn(x;a,b,c;q)
big q-Jacobi polynomial; (18.27.5)
pn(x;a,b,c,d| q)
Askey–Wilson polynomial; (18.28.1)
ph
phase; (1.9.7)
ϕ(z)
Airy phase function; 9.8(i)
Φ(z)=12erfc(-z/2)
alternative notation for the complementary error function; 7.1
(with erfcz: complementary error function)
ϕ(n)
Euler’s totient; (27.2.7)
ϕν(x)
phase of derivatives of Bessel functions; (10.18.3)
Φ1(t;x)
fold catastrophe; (36.2.1)
Φ2(t;x)
cusp catastrophe; (36.2.1)
Φ3(t;x)
swallowtail catastrophe; (36.2.1)
ΦK(t;x)
cuspoid catastrophe; (36.2.1)
ϕk(n)
sum of powers of integers relatively prime to n; (27.2.6)
ϕλ(α,β)(t)
Jacobi function; (15.9.11)
ϕ(z,s)=Lis(z)
notation used by (Truesdell, 1945); 25.12(ii)
(with Lis(z): polylogarithm)
φn,m(z,q)
combined theta function; 20.11(v)
Φ(E)(s,t;x)
elliptic umbilic catastrophe; (36.2.2)
Φ(H)(s,t;x)
hyperbolic umbilic catastrophe; (36.2.3)
ϕ(ρ,β;z)
generalized Bessel function; (10.46.1)
Φ(a;b;z)=M(a,b,z)
notation used by Humbert (1920); 13.1
(with M(a,b,z): Kummer confluent hypergeometric function)
Φ(z,s,a)
Lerch’s transcendent; (25.14.1)
Φ(1)(a;b,b;c;x,y)
first q-Appell function; (17.4.5)
Φ(2)(a;b,b;c,c;x,y)
second q-Appell function; (17.4.6)
Φ(3)(a,a;b,b;c;x,y)
third q-Appell function; (17.4.7)
Φ(4)(a;b;c,c;x,y)
fourth q-Appell function; (17.4.8)
ϕsr+1(a0,a1,,arb1,b2,,bs;q,z)
basic hypergeometric (or q-hypergeometric) function; (17.4.1)
ϕsr+1(a0,a1,,ar;b1,b2,,bs;q,z)
basic hypergeometric (or q-hypergeometric) function; (17.4.1)
π
set of plane partitions; 26.12(i)
Π(z-1)=Γ(z)
notation used by Gauss; 5.1
(with Γ(z): gamma function)
π(x)
number of primes not exceeding x; (27.2.2)
Πm(a)=Γm(a+12(m+1))
notation used by Herz (1955, p. 480); 35.1
(with Γm(a): multivariate gamma function)
Π(n\α)=Π(α2,k)
notation used by Abramowitz and Stegun (1964, Chapter 17); 19.1
(with Π(α2,k): Legendre’s complete elliptic integral of the third kind)
Π(α2,k)
Legendre’s complete elliptic integral of the third kind; (19.2.8)
Π(ϕ,ν,k)=Π(ϕ,α2,k)
notation used by Erdélyi et al. (1953b, Chapter 13); 19.1
(with Π(ϕ,α2,k): Legendre’s incomplete elliptic integral of the third kind)
Π(ϕ,α2,k)
Legendre’s incomplete elliptic integral of the third kind; (19.2.7)
pp(n)
number of plane partitions of n; 26.12(i)
pq(z,k)
generic Jacobian elliptic function; (22.2.10)
Psnm(z,γ2)
spheroidal wave function of complex argument; 30.6
Psnm(z,γ2)=Psnm(z,γ2)
notation used by Meixner and Schäfke (1954) for the spheroidal wave function of complex argument; 30.1
(with Psnm(z,γ2): spheroidal wave function of complex argument)
Psnm(x,γ2)
spheroidal wave function of the first kind; 30.4(i)
psnm(x,γ2)=Psnm(x,γ2)
notation used by Meixner and Schäfke (1954) for the spheroidal wave function of the first kind; 30.1
(with Psnm(x,γ2): spheroidal wave function of the first kind)
ψ(x)
Chebyshev ψ-function; (25.16.1)
Ψ(z)=ψ(z)
notation used by Davis (1933); 5.1
(with ψ(z): psi (or digamma) function)
Ψ(z-1)=ψ(z)
notation used by Gauss, Jahnke and Emde (1945); 5.1
(with ψ(z): psi (or digamma) function)
ψ(z)
psi (or digamma) function; (5.2.2)
Ψ2(x)
Pearcey integral; (36.2.14)
ΨK(x)
canonical integral; (36.2.4)
ψ(n)(z)
polygamma functions; 5.15
Ψ(E)(x)
canonical integral; (36.2.6)
Ψ(H)(x)
canonical integral; (36.2.8)
Ψ(U)(x)
canonical umbilic integral; (36.2.5)
Ψ3(x;k)
swallowtail canonical integral function; (36.2.10)
Ψ3(x;k)
swallowtail canonical integral function; 36.3(i)
ΨK(x;k)
diffraction catastrophe; (36.2.10)
Ψ(E)(x;k)
elliptic umbilic canonical integral function; (36.2.11)
Ψ(E)(x;k)
elliptic umbilic canonical integral function; 36.3(i)
Ψ(E)(x;k)
elliptic umbilic canonical integral function; 36.3(ii)
Ψ(H)(x;k)
hyperbolic umbilic canonical integral function; (36.2.11)
Ψ(H)(x;k)
hyperbolic umbilic canonical integral function; 36.3(i)
Ψ(H)(x;k)
hyperbolic umbilic canonical integral function; 36.3(ii)
Ψ(U)(x;k)
umbilic canonical integral function; (36.2.11)
Ψ(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)
Ψ(a;b;T)
confluent hypergeometric function of matrix argument (second kind); 35.1
Ψ(a;b;T)
confluent hypergeometric function of matrix argument (second kind); (35.6.2)
ψsr(a1,a2,,arb1,b2,,bs;q,z)
bilateral basic hypergeometric (or bilateral q-hypergeometric) function; (17.4.3)
ψsr(a1,a2,,ar;b1,b2,,bs;q,z)
bilateral basic hypergeometric (or bilateral q-hypergeometric) function; (17.4.3)