Introductions to HarmonicNumber2

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1 Itroductios to HarmoicNumber2 Itroductio to the differetiated gamma fuctios Geeral Almost simultaeously with the developmet of the mathematical theory of factorials, biomials, ad gamma fuctios i the 8th cetury, some mathematicias itroduced ad studied related special fuctios that are basically derivatives of the gamma fuctio. These fuctios appeared i coefficiets of the series expasios of the solutios i the logarithmic cases of some importat differetial euatios. They appear i the Bessel differetial euatio for example. So fuctios i this group are called the differetiated gamma fuctios. The harmoic umbers H H, H 2 2, H 3 2 3, H 4 2 3, for iteger have a 4 very log history. The famous Pythagoras of Samos ( B.C.) was the first to ecouter the harmoic series k k i coectio with strig vibratios ad his special iterest i music. Richard Suiseth (4th cetury) ad Nicole d Oresme (350) studied the harmoic series ad discovered that it diverges. Pietro Megoli (647) proved the divergece of the harmoic series. Nicolaus Mercator (668) studied the harmoic series correspodig to the series of log z ad Jacob Beroulli (689) agai proved the divergece of the harmoic series. The harmoic umbers H with iteger also appeared i a article of G. W. Leibiz (673). I his famous work, J. Stirlig (730) ot oly foud the asymptotic formula for factorial, but used the digamma psi fuctio Ψz (related to the harmoic umbers), which is eual to the derivative of the logarithm from the gamma fuctio (Ψz logz z). Later L. Euler (740) also used harmoic umbers ad itroduced the geeralized harmoic umbers H r. The digamma fuctio Ψz ad its derivatives Ψ z of positive iteger orders were widely used i the research of A. M. Legedre (809), S. Poisso (8), C. F. Gauss (80), ad others. M. A. Ster (847) proved the covergece of the Stirlig series for the digamma fuctio: k k j z j Ψz k k. k

2 2 At the ed of the 20th cetury, mathematicias bega to ivestigate extedig the fuctio Ψ z to all complex values of (B. Ross (974), N. Grossma (976)). R. W. Gosper (997) defied ad studied the cases 2 ad 3. V. S. Adamchik (998) suggested the defiitio Ψ z for complex usig Liouville's fractioal itegratio operator. A atural extesio of Ψ z for the complex order Ν was recetly suggested by O. I. Marichev (200) durig the developmet of subsectios with fractioal itegro-differetiatio for the Wolfram Fuctios website ad the techical computig system Mathematica: Ψ Ν z z Ν logz ΨΝ z Ν Ν z Ν 2F k 2, 2; 2 Ν; z k. k Defiitios of the differetiated gamma fuctios The digamma fuctio Ψz, polygamma fuctio Ψ Ν z, harmoic umber H z, ad geeralized harmoic umber H r z are defied by the followig formulas (the first formula is a geeral defiitio for complex argumets ad the secod formula is for positive iteger argumets): Ψz k k k z Ψ k ;. k Here is the Euler gamma costat : Ψ Ν z z Ν logz ΨΝ z Ν Ν z Ν 2F k 2, 2; 2 Ν; z k. k Remark: This formula presets the (ot uiue) cotiuatio of the classical defiitio of Ψ Ν z from positive iteger values of Ν to its arbitrary complex values. For positive iteger ad arbitrary complex z, r, the followig defiitios are commoly used: Ψ z ; k z Rez 0 H z Ψz H ; k k H z r Ζr Ζr, z H r k ;. r k

3 3 The previous defiitio for H z r uses the Mathematica defiitio for the Hurwitz zeta fuctio Ζr, z. Brach cuts ad related properties are thus iherited from Ζr, z. The previous fuctios are itercoected ad belog to the differetiated gamma fuctios group. These fuctios are widely used i the coefficiets of series expasios for may mathematical fuctios (especially the so-called logarithmic cases). A uick look at the differetiated gamma fuctios Here is a uick look at the graphics for the differetiated gamma fuctios alog the real axis. Ψ 0 x Ψ x Ψ 2 x Ψ 3 x Ψ 4 x x H k x H k x H k x H k x H k Coectios withi the group of differetiated gamma fuctios ad with other fuctio groups Represetatios through more geeral fuctios The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r are particular cases of the more geeral hypergeometric ad Meijer G fuctios. Although the argumets of these fuctios do ot deped o the variable z, it is icluded i their parameters. For example, Ψz, Ψ Ν z, H z, ad H z r ca be represeted through geeralized hypergeometric fuctios p F p by the followig formulas: Ψz z 3 F 2,, 2 z; 2, 2; Ψ z z 2 3F 2,, 2 z; 2, 2; 2 2 z 4 F 3,,, 2 z; 2, 2, 2; Π2 6 Ψ z z 2 F, a, a 2,, a ; a, a 2,, a ; ; a a 2 a z H z z 3 F 2,, z; 2, 2;

4 4 H 2 z 2 z 4 F 3,,, z; 2, 2, 2; z 2 3F 2,, z; 2, 2; 2 H r z r F r, a, a 2,, a r ; a, a 2,, a r ; z r rf r, b, b 2,, b r ; b, b 2,, b r ; ; a a 2 a r b b 2 b r z r. The aforemetioed geeral formulas ca be rewritte usig the classical Meijer G fuctios as follows: Ψz z G 3,3,3 0, 0, z 0,, Ψ z,2 G 2,2 0, z,, z 0, z,, z ; H z z G 3,3,3 0, 0, z 0,, H r,r z G r,r 0, a,, a r,r G 0, a,, a r,r r a a 2 a r b b 2 b r z r. Represetatios through related euivalet fuctios 0, b,, b r 0, b,, b r ; The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r ca be represeted through derivatives of the gamma fuctio z: Ψz z z z Ψ z z z z z ; z H z z z H z r r r r r z z z z z0 r z r z z z ; r. The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H r z ca also be represeted through derivatives of the logarithm of the gamma fuctio logz: Ψz logz z Ψ z logz ; z logz H z z

5 5 H r z r r r logz z0 r logz ; r. z r z r The fuctios Ψ z ad H r z are itimately related to the Hurwitz zeta fuctio Ζs, a ad the Beroulli polyomials ad umbers B m z, B m by the formulas: Ψ z Ζ, z ; Rez 0 H r z Ζr Ζr, z H m B m B m m ; m. Represetatios through other differetiated gamma fuctios The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r are itercoected through the followig formulas: Ψz Ψ 0 z Ψz H z Ψ z Ψz ; z H H H z Ψz H r z r r Ψr Ψ r z ; r. The best-kow properties ad formulas for differetiated gamma fuctios Real values for real argumets For real values of the argumet z ad oegative iteger, the differetiated gamma fuctios Ψz, Ψ z, H z, ad H z are real (or ifiity). The fuctio H z is real (or ifiity) for real values of argumet z ad iteger. Simple values at zero The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H r z have simple values for zero argumets: Ψ0 Ψ 0 0 Ψ Ν 0 0 ; ReΝ Ψ Ν 0 ; ReΝ Ψ 0 z Ψz H 0 0

6 6 H r 0 0 H 0 z z. Values at fixed poits The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H r z with ratioal argumets ca sometimes be evaluated through classical costats ad logarithms, for example: Ψ Ψ 2 2 log4 Ψ0 Ψ 2 log4 Ψ Ψ 3 2 2log4 Ψ 8 C Π2 4 H H 2 H 0 0 log4 H 2 2 log4 H H log4. Specific values for specialized variables The previous relatios are particular cases of the followig geeral formulas: Ψ ; k k Ψ ; 2 2 Ψ 2 k k log4 ; k k

7 Ψ 2 k k log4 ; k k Ψ p p k 2 k 2 cos 2 Π p k log si Π k Π 2 cot Π p log2 ; p p Ψ p k p 2 k 2 cos 2 Π p k log si Π k Π 2 cot Π p log2 ; p p H ; k k H ; 2 2 H 2 k k k 2 2 H 2 k k k k 2 2 k 2 2 log4 ; log4 ; H p p k 2 k 2 cos 2 Π p k log si Π k Π 2 cot Π p log2 ; p p H p 2 k p 2 k 2 cos 2 Π p k log si Π k Π 2 cot Π p log2 ; p p. The differetiated gamma fuctios Ψ Ν z ad H r z with iteger parameters Ν ad r have the followig represetatios: Ψ z logz Ψ 0 z Ψz Ψ z Ψz ; z H 2 z z z 2 z 6 H z z z 2 H z 0 z H z H z

8 8 H 0 r 0 H r H 2 r 2 r H 3 r 2 r 3 r. Aalyticity The digamma fuctio Ψz ad the harmoic umber H z are defied for all complex values of the variable z. The fuctios Ψ t z ad H z r are aalytical fuctios of r ad z over the whole complex r- ad z-plaes. For fixed z, the geeralized harmoic umber H z r is a etire fuctio of r. Poles ad essetial sigularities The differetiated gamma fuctios Ψz ad H z have a ifiite set of sigular poits z k, where k for Ψz ad k for H z. These poits are the simple poles with residues. The poit z is the accumulatio poit of poles for the fuctios Ψz, H z, ad Ψ Ν z (with fixed oegative iteger Ν), which meas that is a essetial sigular poit. For fixed oegative iteger Ν, the fuctio Ψ Ν z has a ifiite set of sigular poits: z m ; Ν 0 m are the simple poles with residues ; ad z m ; Ν 0 m are the poles of order Ν with residues 0. For fixed z, the fuctio Ψ Ν z does ot have poles ad the fuctio H z r has oly oe sigular poit at r, which is a essetial sigular poit. Brach poits ad brach cuts The fuctios Ψz ad H z do ot have brach poits ad brach cuts. For iteger Ν, the fuctio Ψ Ν z does ot have brach poits ad brach cuts. For fixed oiteger Ν, the fuctio Ψ Ν z has two sigular brach poits z 0 ad z, ad it is a siglevalued fuctio o the z-plae cut alog the iterval, 0, where it is cotiuous from above: lim Ψ Ν x Ε Ψ Ν x ; x 0 Ε0 lim Ψ Ν x Ε 2 Π Ν Ε0 2 Π Ψ Ν xν x Ν ; x 0. For fixed z, the fuctios Ψ Ν z ad H r z do ot have brach poits ad brach cuts. Periodicity The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H r z do ot have periodicity. Parity ad symmetry

9 9 The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r have mirror symmetry: Ψz Ψz Ψ Ν z Ψ Ν z H z H z H z r H z r. Series represetatios The differetiated gamma fuctios Ψz, H z, ad H z r have the followig series expasios ear regular poits: Ψz Ψz 0 Ζ2, z 0 z z 0 Ζ3, z 0 z z 0 2 ; z z 0 z 0 z 0 0 Ψz Ψz 0 j Ζj 2, z 0 z z 0 j ; z 0 z 0 0 j 0 j z z 0 j Ψz Ψz 0 j 0 k z 0 j2 ; z 0 z 0 0 H z Π2 z 6 Ζ3 z2 Π4 z3 90 ; z 0 H z j Ζj 2 z j ; z j 0 H z H z0 Ζ2, z 0 z z 0 Ζ3, z 0 z z 0 2 ; z z 0 j z z 0 j H z H z0 j 0 k z 0 j2 H z r r Ζr z 2 r r Ζr 2 z2 ; Rer z j r j Ζj r z j H r z j j ; Rer z H z r H z k logk z k z logk k r 2 k log 2 k z k z log2 k k r 2 ; r. Near sigular poits, the differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H r z ca be expaded through the followig series: Ψz z Π2 z 6 Ζ3 z2 Π4 z3 90 ; z 0

10 0 Ψz z j 0 j Ζj 2 z j ; z Ψz Π2 Ψ z 3 Ζ2, z Ζ3, z 2 ; z Ψz z Ψ k Ψ k Ζk Ζk, z k ; z k Ψ Ν z Ν exp z, z Ν zν Ν Ψ Ν z Ν exp z, z Ν zν Π2 zν 6 2 Ν Ν zν k 2 Ζ3 z Π 2 2 Ν k 2 2F, 2; 2 Ν; z k Ψ z Ψ Ψ z 2 Ψ2 z 2 ; z 0 z 2 Π 2 z 2 ; z 0 ReΝ Ν 3 Ν Ψ z Ψ k z k k z ; z Ψ z H z m m Ψ 2 H 2 m Ψ z m 2 3 H m 3 Ψ 2 z m 2 ; z m m Ψ z Ψ k k k H m z m k ; z m m z m k H z z H Π2 3 Ζ2, z Ζ3, z 2 ; z H z z H k Ψ k Ζk Ζk, z k ; z k H z r z m r r H m r r r k Here is the Euler gamma costat k r H kr m Ψ kr z m k ; z m m r. k Except for the geeralized power series, there are other types of series through which differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r ca be represeted, for example: Ψz z z k k z

11 Ψ z ; k z H z z k k z H z r k k r k z r ; Rer. Asymptotic series expasios The asymptotic behavior of the differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r ca be described by the followig formulas (oly the mai terms of the asymptotic expasios are give): Ψz logz 2 z Ψ z z H z logz 2 z 2 z 2 O z 2 ; Argz Π z 2 z 2 z 2 O z 2 ; Argz Π z 2 z 2 O z 2 ; Argz Π z H z r r Ψ r r zr r O z ; Argz Π r z. Itegral represetatios The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H r z ca also be represeted through the followig euivalet itegrals: t z Ψz t ; Rez 0 0 t Ψz 0 t t t z t t ; Rez 0 Ψz 0 t z t Ψ Ν z 0 t t ; Rez 0 t zν t Ν tν z t QΝ, 0, t z t Ψ z 0 t t z t z H z t ; Rez 0 t t t ; Rez 0 zν ; Rez 0 Ν

12 2 H z 0 t t t z t t ; Rez H z 0 t z t t t ; Rez H r z r r Ψ r t r t z t ; r Rez 0 t H r z r r t z log r t t ; Rez r. 0 t Trasformatios The followig formulas describe some of the trasformatios that chage the differetiated gamma fuctios ito themselves: Ψz Ψz Π cotπ z z Ψz Ψz z Ψz Ψz ; z k Ψz Ψz z Ψz Ψz ; z k Ψ z Ψ z z Π cotπ z ; z Ψ z Ψ z z ; Ψ z m Ψ z m ; z k Ψ z Ψ z z ; Ψ z m Ψ z m k ; z k H z H z Π cotπ z z H z H z z

13 3 H z H z ; k z k H z H z z H z H z ; z k r r r H r z r z r 2 2 Π r B r Π r, r H z Π r r cotz Π ; r z r r H z r H z H r z z r H z r k ; r k z r H z H z r z r r H z H z r ;. r z k Trasformatios with argumets that are iteger multiples take the followig forms: Ψ2 z log2 2 Ψ z 2 Ψz m Ψm z logm m Ψ z k m ; m Ψ 2 z 2 Ψ z Ψ z 2 ; m Ψ m z m Ψ z k m ; m H 2 z 2 H z 2 H z log2 m H m z m H k logm ; m z m H r 2 z 2 r r H z r H z 2 2 r z r 2 r Ζr ; Rez 0 r m r H m z m r H k m r Ζr ; Rez 0 m. z m

14 4 The followig trasformatios represet summatio theorems: Ψz Ψ z 2 2 Ψ2 z 2 log2 Ψ z Ψ z 2 2 Ψ 2 z ; H z H 2 H z 2 z 2 log2 2 r H z r H z 2 Idetities 2 r r H 2 z 2 2 r Ζr ; Rez 2. The differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r satisfy the followig recurrece idetities: Ψz Ψz z Ψz Ψz z Ψ z Ψ z z ; Ψ z Ψ z z ; H z H z z H z H z z H r r z H z z r H r r z H z z. r The previous formulas ca be geeralized to the followig recurrece idetities with a jump of legth : Ψz Ψz ; z k Ψz Ψz ; z k k Ψ z Ψ z m m ; z k

15 5 m Ψ z Ψ z m ; k z k H z H z ; z k k H z H z ; z k H r r z H z H r r z H z k ; r z k ;. r z k Represetatios of derivatives The derivatives of the differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H z r have rather simple represetatios: Ψz Ψ z z Ψ Ν z Ψ Ν z z H z z Π2 6 H z 2 H z r z H r r r Ζr H z r logk ;. k 2 k r The correspodig symbolic th -order derivatives of all the differetiated gamma fuctios Ψz, Ψ Ν z, H z, ad H r z ca be expressed by the followig formulas: Ψz Ψ z ; z m Ψ Ν z Ψ Νm z ; m z m H z z H z Ζ ; H z r z Ζr r H z r Ζ r ;

16 6 r H m m log k ; m. r k r k 2 Applicatios of differetiated gamma fuctios Applicatios of differetiated gamma fuctios iclude discrete mathematics, umber theory, ad calculus.

17 7 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a key to the otatios used here, see Please cite this documet by referrig to the fuctios.wolfram.com page from which it was dowloaded, for example: To refer to a particular formula, cite fuctios.wolfram.com followed by the citatio umber. e.g.: This documet is curretly i a prelimiary form. If you have commets or suggestios, please commets@fuctios.wolfram.com , Wolfram Research, Ic.