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11: 5.15 Polygamma Functions
The functions ψ ( n ) ( z ) , n = 1 , 2 , , are called the polygamma functions. In particular, ψ ( z ) is the trigamma function; ψ ′′ , ψ ( 3 ) , ψ ( 4 ) are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. …
5.15.1 ψ ( z ) = k = 0 1 ( k + z ) 2 , z 0 , - 1 , - 2 , ,
5.15.5 ψ ( n ) ( z + 1 ) = ψ ( n ) ( z ) + ( - 1 ) n n ! z - n - 1 ,
12: 5.10 Continued Fractions
§5.10 Continued Fractions
13: 5.19 Mathematical Applications
§5.19(i) Summation of Rational Functions
5.19.3 S = ψ ( 1 2 ) - 2 ψ ( 2 3 ) - γ = 3 ln 3 - 2 ln 2 - 1 3 π 3 .
14: 25.16 Mathematical Applications
In studying the distribution of primes p x , Chebyshev (1851) introduced a function ψ ( x ) (not to be confused with the digamma function used elsewhere in this chapter), given by
25.16.1 ψ ( x ) = m = 1 p m x ln p ,
25.16.2 ψ ( x ) = x - ζ ( 0 ) ζ ( 0 ) - ρ x ρ ρ + o ( 1 ) , x ,
25.16.3 ψ ( x ) = x + o ( x ) , x .
25.16.4 ψ ( x ) = x + O ( x 1 2 + ϵ ) , x ,
15: 5.6 Inequalities
Kershaw’s Inequality
5.6.5 exp ( ( 1 - s ) ψ ( x + s 1 / 2 ) ) Γ ( x + 1 ) Γ ( x + s ) exp ( ( 1 - s ) ψ ( x + 1 2 ( s + 1 ) ) ) , 0 < s < 1 .
16: 36.6 Scaling Relations
Ψ K ( x ; k ) = k β K Ψ K ( y ( k ) ) ,
Ψ ( U ) ( x ; k ) = k β ( U ) Ψ ( U ) ( y ( U ) ( k ) ) ,
Indices for k -Scaling of Magnitude of Ψ K or Ψ ( U ) (Singularity Index)
17: 36.3 Visualizations of Canonical Integrals
Figure 36.3.2: Modulus of swallowtail canonical integral function | Ψ 3 ( x , y , 3 ) | . …
Figure 36.3.3: Modulus of swallowtail canonical integral function | Ψ 3 ( x , y , 0 ) | . …
Figure 36.3.4: Modulus of swallowtail canonical integral function | Ψ 3 ( x , y , - 3 ) | . …
Figure 36.3.5: Modulus of swallowtail canonical integral function | Ψ 3 ( x , y , - 7.5 ) | . …
Figure 36.3.6: Modulus of elliptic umbilic canonical integral function | Ψ ( E ) ( x , y , 0 ) | . …
18: 36.2 Catastrophes and Canonical Integrals
36.2.5 Ψ ( U ) ( x ) = - - exp ( i Φ ( U ) ( s , t ; x ) ) d s d t , U = E , H .
36.2.11 Ψ ( U ) ( x ; k ) = k - - exp ( i k Φ ( U ) ( s , t ; x ) ) d s d t , U = E , H ; k > 0 .
Ψ 1 is related to the Airy function9.2):
36.2.13 Ψ 1 ( x ) = 2 π 3 1 / 3 Ai ( x 3 1 / 3 ) .
36.2.15 Ψ K ( 0 ) = 2 K + 2 Γ ( 1 K + 2 ) { exp ( i π 2 ( K + 2 ) ) , K  even, cos ( π 2 ( K + 2 ) ) , K  odd .
19: 17.1 Special Notation
The main functions treated in this chapter are the basic hypergeometric (or q -hypergeometric) function ϕ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , the bilateral basic hypergeometric (or bilateral q -hypergeometric) function ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , and the q -analogs of the Appell functions Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) , Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) , Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) , and Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) . …
20: 14.11 Derivatives with Respect to Degree or Order
14.11.3 A ν μ ( x ) = sin ( ν π ) ( 1 + x 1 - x ) μ / 2 k = 0 ( 1 2 - 1 2 x ) k Γ ( k - ν ) Γ ( k + ν + 1 ) k ! Γ ( k - μ + 1 ) ( ψ ( k + ν + 1 ) - ψ ( k - ν ) ) .