Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

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1 PolyGamma Notations Traditional name Digamma function Traditional notation Ψz Mathematica StandardForm notation PolyGammaz Primary definition Ψz k k k z Specific values Specialized values n Ψn ; n k k Ψn ; n Ψ n n 4 4 k Π log8 ; n Ψ n 4 n 4 4 k 3 Π log8 ; n Ψ n n k 9 log3 3 Π ; n 6

2 Ψ n 3 n 3 3 k 2 9 log3 3 Π ; n n 2 n 2 Ψ n 2 k k log4 ; n k n k n 2 n 2 Ψ 2 n k k log4 ; n k n k Ψ n 2 n k 2 3 Π 9log3 ; n Ψ 2 n 3 n 3 3 k 3 Π log9 683 ; n Ψ n 3 n k 3 Π log8 ; n Ψ 3 n 4 n 4 4 k Π log8 ; n Ψ n p n p k 2 k 2 cos 2 Π p k log sin Π k Π 2 cot Π p log2 ; n p p n Ψ p n k p 2 k 2 cos 2 Π p k log sin Π k Π 2 cot Π p log2 ; n p p Ψ p 2 k 2 cos 2 Π p k log sin Π k 2 Π cot Π p log2 ; p p Values at fixed points Ψ3

3 3 Ψ log Ψ Ψ log Ψ Ψ 2 2 log Ψ Ψ 2 log Ψ Ψ 3 2 2log Ψ Ψ log4 Ψ Values at infinities Ψ Ψ Ψ Ψ Ψ

4 4 General characteristics Domain and analyticity Ψz is an analytical function of z which is defined over the whole complex z-plane with the exception of countably many points z k ; k zψz Symmetries and periodicities Mirror symmetry Ψz Ψz Periodicity No periodicity Poles and essential singularities The function Ψz has an infinite set of singular points: a) z k ; k, are the simple poles with residues ; b) z is the point of convergence of poles, which is an essential singular point ing z Ψz k, ; k,, res z Ψzk ; k Branch points The function Ψz does not have branch points z Ψz Branch cuts The function Ψz does not have branch cuts z Ψz Series representations Generalized power series

5 5 Expansions at z 0 For the function itself Ψz z Π2 z 6 Ζ3 Π4 z3 z2 ; z Ψz z Π2 z 6 Ζ3 z Π4 z3 90 Oz4 Ψz z j Ζj 2 z j ; z j Ψz z j z j ; z j2 j 0 k Ψz Oz z Expansions at z z 0 ; z 0 n For the function itself Ψz Ψz 0 Ζ2, z 0 z z 0 Ζ3, z 0 z z 0 2 ; z z 0 z 0 z Ψz Ψz 0 Ζ2, z 0 z z 0 Ζ3, z 0 z z 0 2 Oz z 0 3 ; z 0 z Ψz Ψz 0 j Ζj 2, z 0 z z 0 j ; z 0 z 0 0 j j z z 0 j Ψz Ψz 0 j 0 k z 0 j2 ; z 0 z Ψz Ψz 0 Ζ2, z 0 z z 0 Oz z 0 ; z 0 z 0 0 Expansions at z n For the function itself

6 Ψz Π2 Ψn z n 3 Ζ2, n z n Ζ3, n z n2 Π4 45 Ζ4, n z n3 ; z n n Ψz Π2 Ψn z n 3 Ζ2, n z n Ζ3, n z n2 Π4 45 Ζ4, n z n3 Oz n 4 ; n Ψz z n Ψn k Ψ k Ζk Ζk, n z n k ; n k Ψz Ψn Oz n ; n z n Expansions of Ψz Ε at Ε 0 ; z n For the function itself Ψz Ε Ψz Ζ2, z Ε Ζ3, z Ε 2 ; Ε 0 z z Ψz Ε Ψz Ζ2, z Ε Ζ3, z Ε 2 OΕ 3 ; z z Ψz Ε Ψz j Ζj 2, z Ε j ; z z 0 j Ψz Ε Ψz OΕ ; z z 0 Expansions of Ψn Ε at Ε 0 For the function itself Ψn Ε Π2 Ψn Ε 3 Ζ2, n Ε 2 Ψ2 n Ε 2 ; Ε 0 n Ψn Ε Π2 Ψn Ε 3 Ζ2, n Ε 2 Ψ2 n Ε 2 OΕ 3 ; n Ψn Ε Ε Ψn k Ψ k Ζk Ζk, n Ε k ; Ε 0 n k Ψn Ε OΕ ; n Ε

7 7 Asymptotic series expansions Ψz logz 2 z k B 2 k ; argz Π z 2 k 2 k z Ψz logz 2 z k B 2 k Π cotπ z argz 2 k z 2 k Π ; z z 0 z Ψz logz 2 z 2 z 2 O z 2 ; argz Π z Ψz logz 2 z Residue representations 2 z 2 O z 2 Π cotπ z argz Π ; z z 0 z Ψz z res s j 0 s 2 2 s z s s j 2 s 2 Other series representations Ψz z z z 2 z 2 3 z Ψz z z k k z Integral representations On the real axis Of the direct function t z Ψz t ; Rez 0 0 t t t z Ψz 0 t t t ; Rez Ψz 0 t z t t t ; Rez 0

8 Ψn P n t 2 t ; n t Ψz 0 A. Radovi x z x logx x ; Rez 0 Contour integral representations Ψz 2 Π z s s s 2 z s s s 2 s 2 s Γ Ψz 2 Π z s s s 2 z s s s ; 0 Γ Γ 2 s 2 s Limit representations Ψz lim n n logn z k Ψz lim Ζs, z s s Generating functions t log t Ψn t n t ; n Transformations Transformations and argument simplifications Argument involving basic arithmetic operations Ψ z Π cotπ z Ψz Ψz Ψz Π cotπ z z Ψz Ψz z

9 Ψz Ψz z n Ψz n Ψz ; n z k n Ψz n Ψz ; n z k Multiple arguments Argument involving numeric multiples of variable Ψ2 z log2 2 Ψ z 2 Ψz Ψ3 z 3 Ψz Ψ z 3 Ψ z 2 3 log3 Argument involving symbolic multiples of variable Ψm z logm m m Ψ z k m ; m Products, sums, and powers of the direct function Sums of the direct function Ψz Ψ z 2 2 Ψ2 z 2 log2 Identities Recurrence identities Consecutive neighbors Ψz Ψz z Ψz Ψz z Distant neighbors

10 n Ψz Ψz n ; n z k n Ψz Ψz n ; n k z k Functional identities Relations of special kind Ψz Ψz Π cotπ z z Complex characteristics Real part Ψ 0 x 2 x y 2 ReΨx y RootSum x 2 2 x y 2 2 &, & x x y ReΨx y Ψx y Ψx y 2 Imaginary part ImΨx y Ψx y Ψx y 2 Differentiation Low-order differentiation Ψz Ψ z z Ψz Ψ 2 z z 2 Symbolic differentiation n Ψz Ψ n z ; n z n

11 Fractional integro-differentiation Α Ψz Ψ Α z z Α Α Ψz z Α ΝΑ exp z, z Α zα Α zα k k 2 2 F, 2; 2 Α; z k Integration Indefinite integration Involving only one direct function Ψz z logz Involving one direct function and elementary functions Involving power function z Α Ψz z zα Α zα Α zα Α n z n Ψz z j n j j z nj Ψ j z ; n j 0 z 3F 2, 2, Α ; 2, Α 2; k 2 k Definite integration Involving the direct function n 0 zψt z t n t n Ψ n z k n k k z kn j j k j Ψ j ; n j 0 Summation Finite summation n Ψk n Ψn ; n k

12 2 k Ψ k 2 Π p k 2 Π p log ; p p m i p m i k a k i i n i k b k i Ψi Ψi n Ψm i p n m k n a k k m p k n b k pmn z n m b k n k a k k m m b k n k m a k k p m n m k n n j 0 m Ψ i a k Ψ i b k Ψi a k k m a k k m m b k n k n pmn z mn p2f 2,, m a,, m a p ; m n 2, m 2, m b,, m b ; z n j m k j n b k n k j n a k p jm n j k n n j a k k m j n b k m b i b i n i m k b i b k k i p i m b i k n mnp z j b i b k n k a k b i b i a k k m b i b k k m Ψi b k z i mnp z nb i p2f 2, m b i, a b i,, a p b i ; n b i 2, 2 b i, b b i,, b b i ; z log pmn z p F m, a,, a p ; n, b,, b ; z n p n m k n k a k k m m b k n k a k b k G m2,n p,2 pmn z a,, a n, m, a n,, a p 0, n, b,, b m, b m,, b ; n m j,k,j,k j k j m k m b j b k

13 3 {k, n +, p}] + Sum[PolyGamma[ b[[k]] + i], {k, m +, }] + PolyGamma[n + i + ] + PolyGamma[i + ] - PolyGamma[m + - i]), {i, 0, m}]/ ((((-)^n*n!*m!*product[gamma[ - a[[k]]], {k, n +, p}]*product[gamma[b[[k]]], {k, m +, }])/ (Product[Gamma[ - b[[k]]], {k,, m}]* Product[Gamma[a[[k]]], {k,, n}]))* MeijerG[{Table[ - a[[r]], {r,, n}], Join[{m + }, Table[ - a[[r]], {r, + n, p}]]}, {Join[{0, -n}, Table[ - b[[r]], {r,, m}]], Table[ - b[[r]], {r, + m, }]}, (-)^(p - n - m + )*z] - (((n!*m!*product[gamma[ - a[[k]]], {k, n +, p}]*product[gamma[b[[k]]], {k, m +, }])/(Product[ Gamma[ - b[[k]]], {k,, m}]* Product[Gamma[a[[k]]], {k,, n}]))* (Sum[Gamma[- + b[[i]]]*gamma[-n - + b[[ i]]]*((product[gamma[-b[[k]] + b[[i]]], {k,, i - }]*Product[ Gamma[-b[[k]] + b[[i]]], {k, i +, m}]*product[gamma[ + a[[k]] - b[[i]]], {k,, n}])/(gamma[ m + b[[i]]]*product[gamma[-a[[k]] + b[[i]]], {k, n +, p}]*product[ Gamma[ + b[[k]] - b[[i]]], {k, m +, }]))*((-)^(p - n - m + )*z)^( + n - b[[i]])* HypergeometricPFQ[Join[{, - m - b[[i]]}, Table[ + a[[r]] - b[[i]], {r,, p}]], Join[{2 + n - b[[i]], 2 - b[[i]]}, Table[ + b[[r]] - b[[i]], {r,, }]], z], {i,, m}] + Sum[(((n - j - )!*Product[ Gamma[ + n - b[[k]] - j], {k,, m}]* Product[Gamma[-n + a[[k]] + j], {k,, n}])/((n + m - j)!*product[ Gamma[ + n - a[[k]] - j], {k, n +, p}]*product[gamma[-n + b[[k]] + j], {k, m +, }]))*(((-)^(p - n - m)* z)^j/j!), {j, 0, n - }] + ((Product[Gamma[-b[[k]] - m], {k,, m}]* Product[Gamma[ + a[[k]] + m], {k,,

14 4 n}])/(product[gamma[-a[[k]] - m], {k, n +, p}]*product[gamma[ + b[[k]] + m], {k, m +, }]))* (((-)^n*((-)^(p - n - m - )*z)^(n + m + ))/((n + m + )!*(m + )!))* HypergeometricPFQ[Join[{, }, Table[ + m + a[[r]], {r,, p}]], Join[{2 + n + m, 2 + m}, Table[ + m + b[[r]], {r,, }]], z]))/ ((-)^(p - n - m)*z)^n + Log[(-)^(p - n - m - )*z]* HypergeometricPFQ[Join[{-m}, Table[a[[r]], {r,, p}]], Join[{n + }, Table[b[[r]], {r,, }]], z])} /. {z -> Random[]*Exp[Pi*I*(ii/4)]}, {n, 0, 3}, {m, 0, 3}, {ii, 0, 7}]] Infinite summation p k a k p m n i Ψi Ψi n Ψ i a k Ψ i b k Ψi a k Ψi b k z i i 0 i n i k b k i k n k k k m n log pmn z b k p F 2, a,, a p ;, n, b,, b ; z k G m2,n p,2 pmn z a,, a n, a n,, a p n, 0, b,, b m, b m,, b n p n k n k n p n k n a k k m m b k n k a k k pmn z n n n j m k j n b k n k j n a k mnp z j p j 0 j k n j n a k k m j n b k b k a k k m b k m b k n k a k m csc 2 Π b i n k a k b i i m n Π m k cscπ b i b k k i cscπ b i b k pmn z bi p i k n b i a k pf 2, a b i,, a p b i ; n b i 2, 2 b i, b b i,, b b i ; z ; n j,k,j,k j k j m k m b j b k

15 5 (i!*pochhammer[n +, i]*product[ Pochhammer[bb[[k]], i], {k,, }]))*z^i* (Sum[PolyGamma[ - bb[[k]] - i], {k,, m}] - Sum[PolyGamma[aa[[k]] + i], {k,, n}] - Sum[PolyGamma[ - aa[[k]] - i], {k, n +, p}] + Sum[PolyGamma[bb[[k]] + i], {k, m +, }] + PolyGamma[n + i + ] + PolyGamma[i + ]), {i, 0, 00}]/ ((((-)^n*n!*product[gamma[ - aa[[k]]], {k, n +, p}]*product[gamma[bb[[k]]], {k, m +, }])/ (Product[Gamma[ - bb[[k]]], {k,, m}]* Product[Gamma[aa[[k]]], {k,, n}]))* MeijerG[{Table[ - aa[[r]], {r,, n}], Table[ - aa[[r]], {r, + n, p}]}, {Join[{-n, 0}, Table[ - bb[[r]], {r,, m}]], Table[ - bb[[r]], {r, + m, }]}, (-)^(p - n - m)*z] - (((-)^n*n!*product[gamma[ - aa[[k]]], {k, n +, p}]*product[gamma[bb[[k]]], {k, m +, }])/ (Product[Gamma[ - bb[[k]]], {k,, m}]* Product[Gamma[aa[[k]]], {k,, n}]))* ((-)^n*pi^(m + )* Sum[(Product[Gamma[ + aa[[k]] - bb[[i]]], { k,, n}]/product[gamma[-aa[[k]] + bb[[i]]], {k, n +, p}])* Csc[Pi*bb[[i]]]^2*Product[Csc[Pi* (-bb[[k]] + bb[[i]])], {k,, i - }]* Product[Csc[Pi*(-bb[[k]] + bb[[i]])], {k, i +, m}]*((-)^(p - n - m)*z)^ ( - bb[[i]])* HypergeometricPFQRegularized[Join[{}, Table[ + aa[[r]] - bb[[i]], {r,, p}]], Join[{n bb[[i]], 2 - bb[[i]]}, Table[ + bb[[r]] - bb[[i]], {r,, }]], z], {i,, m}] + Sum[(((n - j - )!*Product[Gamma[ - bb[[k]] + n - j], {k,, m}]* Product[Gamma[aa[[k]] - n + j], {k,, n}])/(product[gamma[ - aa[[k]] + n - j], {k, n +, p}]* Product[Gamma[bb[[k]] - n + j], {k, m +, }]))*

16 6 (((-)^(p - n - m - )*z)^j/j!), {j, 0, n - }]/((-)^(p - n - m)*z)^n) + Log[(-)^(p - n - m)*z]*n!* Product[Gamma[bb[[k]]], {k,, }]* HypergeometricPFQRegularized[Join[{}, Table[aa[[r]], {r,, p}]], Join[{n +, }, Table[bb[[r]], {r,, }]], z])} /. {z -> Random[]*Exp[Pi*I*(ii/4)]}, {n, 0, 3}, {ii, 0, 7}]] Representations through more general functions Through hypergeometric functions Involving p F Ψz z 3 F 2,, 2 z; 2, 2; Through Meijer G Classical cases for the direct function itself Ψz z G,3 3,3 0, 0, z 0,, Through other functions Involving some hypergeometric-type functions Ψz Ψ 0 z Representations through euivalent functions With related functions Ψz z z z Ψz logz z Ψz H z Zeros

17 Ψz k 0 ;.4 z z z z z z z z k History J. Stirling (730) A.-M. Legendre (809) S. Poisson (8) C. F. Gauss (80) M.A. Stern (847) proved convergence of the Stirling series for digamma function

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