Introductions to Pochhammer

Transcription

1 Itroductios to Pochhaer Itroductio to the factorials ad bioials Geeral The factorials ad bioials have a very log history coected with their atural appearace i cobiatorial probles. Such cobiatorial-type probles were ow ad partially solved eve i aciet ties. The first atheatical descriptios of bioial coefficiets arisig fro expasios of a b for, 3, 4, appeared i the wors of Chia Hsie (5), al-karaji (about ), Oar al-khayyai (8), Bhasara Acharya (5), al- Saaw'al (75), Yag Hui (6), Tshu shi Kih (33), Shih Chieh Chu (33), M. Stifel (544), Cardao (545), Scheubel (545), Peletier (549), Tartaglia (556), Carda (57), Stevi (585), Faulhaber (65), Girard (69), Oughtred (63), Briggs (633), Mersee (636), Ferat (636), Wallis (656), Motort (78), ad De Moivre (73). B. Pascal (653) gave a recursio relatio for the bioial, ad I. Newto (676) studied its cases with fractioal arguets. It was ow that the factorial grows very fast. Its growth speed was estiated by J. Stirlig (73) who foud the faous asyptotic forula for the factorial aed after hi. A special role i the history of the factorial ad bioial belogs to L. Euler, who itroduced the gaa fuctio z as the atural extesio of factorial ( ) for oiteger arguets ad used otatios with paretheses for the bioials (774, 78). C. F. Hideburg (779) used ot oly bioials but itroduced ultioials as their geeralizatios. The oder otatio was suggested by C. Krap (88, 86). C. F. Gauss (8) also widely used bioials i his atheatical research, but the oder bioial sybol was itroduced by A. vo Ettighause (86); later Förstea (835) gave the cobiatorial iterpretatio of the bioial coefficiets. A. L. Crelle (83) used a sybol that otates the geeralized factorial a a a a. Later P. E. Appell (88) ascribed the ae Pochhaer sybol for the otatio of this product because it was widely used i the research of L. A. Pochhaer (89). While the double factorial was itroduced log ago, its extesio for coplex arguets was suggested oly several years ago by J. Keiper ad O. I. Marichev (994) durig the ipleetatio of the fuctio Factorial i Matheatica. The classical cobiatorial applicatios of the factorial ad bioial fuctios are the followig: The factorial gives the uber of possible placeets of people o chairs. The bioial gives the uber of possible selectios of ubers fro a larger group of ubers, for istace o a lotto strip.

2 The ultioial ;,,, is the uber of ways of puttig differet objects ito differet boxes with i the th box,,,,. Defiitios of factorials ad bioials The factorial, double factorial, Pochhaer sybol a, bioial coefficiet, ad ultioial coefficiet ;,,, are defied by the followig forulas. The first forula is a geeral defiitio for the coplex arguets, ad the secod oe is for positive iteger arguets: ; cos 4 a a ; a a a a a a ; ; ; ;,,, ; ;,,, ;. Rear about values at special poits: For Α a ad Ν itegers with a ad a, the Pochhaer sybol Α Ν caot be uiquely defied by a liitig procedure based o the previous defiitio because the two variables Α ad Ν ca approach the itegers a ad with a ad a at differet speeds. For such itegers with a, a, the followig defiitio is used: a a a ; a a.

3 3 Siilarly, for Ν, Κ egative itegers with, the bioial coefficiet caot be uiquely defied by a liitig procedure based o the previous defiitio because the two variables Ν, Κ ca approach egative itegers, with at differet speeds. For egative itegers with, the followig defiitio is used: ;. The previous sybols are itercoected ad belog to oe group that ca be called factorials ad bioials. These sybols are widely used i the coefficiets of series expasios for the ajority of atheatical fuctios. A quic loo at the factorials ad bioials Here is a quic loo at the graphics for the factorial the real axis. x x Ad here is a quic view of the bivariate bioial ad Pochhaer fuctios. For positive arguets, both fuctios are free of sigularities. For egative arguets, the fuctios have a coplicated structure with ay sigularities. x y x y x 4 4 y x 4 4 y 4 Coectios withi the group of factorials ad bioials ad with other fuctio groups Represetatios through ore geeral fuctios Two factorials ad are the particular cases of the icoplete gaa fuctio a, z with the secod arguet beig :, ; Re

4 4 cos 4, ; Re. Represetatios through related equivalet fuctios The factorial, double factorial, Pochhaer sybol a, bioial coefficiet, ad ultioial coefficiet ;,,, ca be represeted through the gaa fuctio by the followig forulas: cos 4 a a a a a a a a a a a a ; ; ;,,, ; May of these forulas are used as the ai eleets of the defiitios of ay fuctios. Represetatios through other factorials ad bioials The factorials ad bioials,, a,, ad ;,,, are itercoected by the followig forulas:. 4 cos si 4 cos cos 4

5 5 ; ; a a a a a a ; a, ;, ; ;,,, ; ;,,, ;. The best-ow properties ad forulas for factorials ad bioials Real values for real arguets For real values of arguets, the values of the factorials ad bioials,, a,, ad ;,,, are real (or ifiity). Siple values at zero The factorials ad bioials,, a,, ad ;,,, have siple values for zero arguets:

6 6 ; ;,,, a ; ; si. Values at fixed poits Studets usually lear the followig basic table of values of the factorials ad i special iteger poits: Specific values for specialized variables If variable is a ratioal or iteger uber, the factorials ad ca be represeted by the followig geeral forulas: ;

7 7 p q q p q p q q ; j ; j ; p q ; p q p q q q p p q ; p q p q j ;. j For soe particular values of the variables, the Pochhaer sybol a has the followig eaigs: a a a a a a ; a a a a a a. Soe well-ow forulas for bioial ad ultioial fuctios are: ;

8 8 ; ;,. Aalyticity The factorials ad bioials,, a,, ad ;,,, are defied for all coplex values of their variables. The factorials, bioials, ad ultioials are aalytical fuctios of their variables ad do ot have brach cuts ad brach poits. The fuctios ad do ot have zeros: ;. Therefore, the fuctios ad are etire fuctios with a essetial sigular poit at z. Poles ad essetial sigularities The factorials ad bioials,, a,, ad ;,,, have a essetial sigularity for ifiite values of ay arguet. This sigular poit is also the poit of covergece of the poles (except for ). The fuctio has a ifiite set of sigular poits: ; are the siple poles with residues. The fuctio has a ifiite set of sigular poits: ; are the siple poles with residues. For fixed a, the fuctio a has a ifiite set of sigular poits: a ; are the siple poles with residues a. For fixed, the fuctio a has a ifiite set of sigular poits: a ; are the siple poles with residues ;. For fixed, the fuctio has a ifiite set of sigular poits: j ; j are the siple poles with residues j j j ;. By variable,, (with the other variables fixed) the fuctio ;,,, has a ifiite set of sigular poits: N j ; j are the siple poles with residues j j N r r r r j ; N r r r r j. Periodicity The factorials ad bioials,, a,, ad ;,,, do ot have periodicity. Parity ad syetry

9 9 The factorials ad bioials,, a,, ad ;,,, have irror syetry: a a ;,,, ;,,,. The ultioial ;,,, has perutatio syetry: ;, ;, j ;,,,,, j,, j ;,,, j,,,, ; j j. Series represetatios The factorials,, ad a have the followig series expasios i the regular poits: Ψ Ψ Ψ ; 4 log4 Ψ log si ; a a Ψ a ; a a S a ; b a b a b Ψb Ψb a b Ψb Ψb Ψb Ψb Ψ b Ψ b a b ; a S j j b j a b j ;. The series expasios of ad ear sigular poits are give by the followig forulas: Ψ 6 3 Ψ 3 Ψ ;

10 log Ψ 4 3 log 3 Ψ log8 3 log log64 Ψ 3 Ψ ;. Asyptotic series expasios The asyptotic behavior of the factorials ad bioials,, a,, ad ;,,, ca be described by the followig forulas (oly the ai ters of asyptotic expasio are give). The first is the faous Stirlig's forula: O ; Arg 4 cos O ; Arg a z O a a ; a Arga a a a O ; Arga O ; Arg si O ; Arg ;,,, Itegral represetatios a O ; a Arga. The factorial ad bioial ca also be represeted through the followig itegrals: t t t ; log t t ; Re t t t t ; Re

11 t t t t t t ;. Trasforatios The followig forulas describe soe of the ai types of trasforatios betwee ad aog factorials ad bioials: csc ; cos csc ; ; a a j a j ; a b a b ;. Soe of these trasforatios ca be called additio forulas, for exaple: a b a b ;

12 a b a b ; a a a. Multiple arguet trasforatios are, for exaple: ; si cos 4 ; a a a. The followig trasforatios are for products of the fuctios: csc, cos csc cos cos 4 4 cos cos cos cos cos 4,.

13 3 Idetities The factorials ad ca be defied as the solutios of the followig correspodig fuctioal equatios: f f ; f g g g f f f ; f g g g f. The factorial is the uique ozero solutio of the fuctioal equatio f f that is logarithically covex for all real ; that is, for which log f is a covex fuctio for. The factorials ad bioials,, a,, ad ;,,, satisfy the followig recurrece idetities: a a a a a a a a a a a a a a a a a a a a

14 4 ;,,, l ;,,, j j ;,,, l, l, l,, j j l ;,,, l, l, l,,. The previous forulas ca be geeralized to the followig recurrece idetities with a jup of legth : ; ; ; a a a a a a a a a a a a a a a a a a a a ;,,, l p p;,,, l, l p, l,, j j p ;,,, j p l p p j p;,,, l, l p, l,,.

15 5 The Pochhaer sybol a ad bioial satisfy the followig fuctioal idetities: a a ; a a a a a a ;. Represetatios of derivatives The derivatives of the fuctios,, a,, ad ;,,, have rather siple represetatios that iclude the correspodig fuctios as factors: Ψ log Ψ log si a a a a Ψa Ψa a Ψa Ψ Ψ Ψ Ψ ;,,, ;,,, Ψ Ψ ;. The sybolic derivatives of the th order for factorials ad bioials,, a,, ad ;,,, have uch ore coplicated represetatios, which ca iclude recursive fuctio calls, regularized geeralized hypergeoetric fuctios F, or Stirlig ubers S :

16 6 z R, ; R, z Ψz R, z R,, z R, z a a a a a a a F a, a,, a, ; a, a,, a ; ; a a S a ; si a a a F a, a,, a, ; a, a,, a ; ; j j S j j j ; j j si j a a a u ;,,, u u u s u j F a, a,, a, ; a, a,, a ; s u F ua, a,, a u, s ; a, a,, a u ; ; a a a u s s u s. Applicatios of factorials ad bioials Applicatios of factorials ad bioials iclude cobiatorics, uber theory, discrete atheatics, ad calculus. j ;

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