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Notations P

*ABCDEFGHIJKLMNO♦P♦QRSTUVWXYZ
PI, PII, PIII, PIII, PIV, PV, PVI
Painlevé transcendents; §32.2(i)
P ( z ) = 1 2 erfc ( - 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 { α β γ a 1 b 1 c 1 z a 2 b 2 c 2 }
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
P z ( a ) = γ ( a , z )
notation used by Batchelder (1967, p. 63); §8.1
(with γ(a,z): incomplete gamma function)
P ν ( x ) = P ν 0 ( x )
Ferrers function of the first kind; §14.1
(with Pνμ(x): Ferrers function of the first kind)
P ν ( z ) = P ν 0 ( z )
Legendre function of the first kind; §14.1
(with Pνμ(z): associated Legendre function of the first kind)
P n ( x )
Legendre polynomial; Table 18.3.1
p k ( n )
number of partitions of n into at most k parts; §26.9(i)
p k ( 𝒟 , n )
number of partitions of n into at most k distinct parts; §26.10(i)
P ν μ ( x ) = P ν μ ( x )
notation used by Erdélyi et al. (1953a), Olver (1997b); §14.1
(with Pνμ(x): Ferrers function of the first kind)
P ν μ ( x )
Ferrers function of the first kind; 14.3.1
P ν μ ( z )
associated Legendre function of the first kind; §14.21(i)
P n * ( x )
shifted Legendre polynomial; Table 18.3.1
P ν μ ( x ) = P ν μ ( x )
notation used by Magnus et al. (1966); §14.1
(with Pνμ(x): Ferrers function of the first kind)
𝔓 ν μ ( z ) = P ν μ ( z )
notation used by Magnus et al. (1966); §14.1
(with Pνμ(z): associated Legendre function of the first kind)
P n ( λ ) ( x ) = C n ( λ ) ( x )
notation used by Szegö (1975, §4.7); §18.1(iii)
(with Cn(λ)(x): ultraspherical (or Gegenbauer) polynomial)
P n ( α , β ) ( x )
Jacobi polynomial; Table 18.3.1
P - 1 2 + i 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 !: factorial (as in n!))
P n ( x ; c )
associated Legendre polynomial; 18.30.6
p k ( m , n )
number of partitions of n into at most k parts, each less than or equal to m; §26.9(i)
P n ( α , β ) ( x ; c )
associated Jacobi polynomial; 18.30.4
P n ( λ ) ( x ; ϕ )
Meixner–Pollaczek polynomial; §18.19
P m , n α , β , γ ( x , y )
triangle polynomial; 18.37.7
Π ( n ; ϕ \ α ) = Π ( ϕ , α 2 , 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 ; g 2 , g 3 )
Weierstrass -function; 23.3.8
P n ( λ ) ( x ; a , b )
Pollaczek polynomial; 18.35.4
p n ( x ; a , b ; q )
little q-Jacobi polynomial; 18.27.13
P n ( α , β ) ( x ; c , d ; q )
big q-Jacobi polynomial; 18.27.6
p n ( x ; a , b , a ¯ , 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
ph
phase; 1.9.7
ϕ ( z )
Airy phase function; §9.8(i)
Φ ( z ) = 1 2 erfc ( - z / 2 )
alternative notation for the complementary error function; §7.1
(with erfcz: complementary error function)
ϕ ( n )
Euler’s totient; 27.2.7
ϕ ν ( x )
phase of derivatives of Bessel functions; 10.18.3
ϕ k ( n )
sum of powers of integers relatively prime to n; 27.2.6
ϕ λ ( α , β ) ( t )
Jacobi function; 15.9.11
ϕ ( z , s ) = Li s ( z )
notation used by (Truesdell, 1945); §25.12(ii)
(with Lis(z): polylogarithm)
φ n , m ( z , q )
combined theta function; §20.11(v)
Φ K ( t ; x )
cuspoid catastrophe of codimension K; 36.2.1
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 ; q ; x , y )
first q-Appell function; 17.4.5
Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y )
second q-Appell function; 17.4.6
Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y )
third q-Appell function; 17.4.7
Φ ( 4 ) ( a ; b ; c , c ; q ; x , y )
fourth q-Appell function; 17.4.8
ϕsr+1(a0,,ar;b1,,bs;q,z) or ϕsr+1(a0,,arb1,,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
Π m ( a ) = Γ m ( a + 1 2 ( m + 1 ) )
notation used by Herz (1955, p. 480); §35.1
(with Γm(a): multivariate gamma function)
Π ( n \ α ) = Π ( α 2 , 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 ( α 2 , k )
Legendre’s complete elliptic integral of the third kind; 19.2.8
Π 1 ( ν , k ) = Π ( α 2 , 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)
Π ( ϕ , ν , k ) = Π ( ϕ , α 2 , 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 , α 2 , 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
Ps n m ( z , γ 2 )
spheroidal wave function of complex argument; §30.6
Ps n m ( z , γ 2 ) = Ps n m ( z , γ 2 )
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)
Ps n m ( x , γ 2 )
spheroidal wave function of the first kind; §30.4(i)
ps n m ( x , γ 2 ) = Ps n m ( x , γ 2 )
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
Ψ 2 ( x )
Pearcey integral; 36.2.14
Ψ K ( x )
canonical integral function; 36.2.4
ψ ( n ) ( z )
polygamma functions; §5.15
Ψ ( E ) ( x )
elliptic umbilic canonical integral function; 36.2.5
Ψ ( H ) ( x )
hyperbolic umbilic canonical integral function; 36.2.5
Ψ ( U ) ( x )
umbilic canonical integral function; 36.2.5
Ψ K ( x ; k )
diffraction catastrophe; 36.2.10
Ψ ( E ) ( x ; k )
elliptic umbilic canonical integral function; 36.2.11
Ψ ( H ) ( x ; k )
hyperbolic umbilic canonical integral function; 36.2.11
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); 35.6.2
ψsr(a1,,ar;b1,,bs;q,z) or ψsr(a1,,arb1,,bs;q,z)
bilateral basic hypergeometric (or bilateral q-hypergeometric) function; 17.4.3