Introductions to PartitionsP

Transcription

1 Itroductios to PartitiosP Itroductio to partitios Geeral Iterest i partitios appeared i the 7th cetury whe G. W. Leibiz (669) ivestigated the umber of ways a give positive iteger ca be decomposed ito a sum of smaller itegers. Later, L. Euler (740) also used partitios i his wor. But extesive ivestigatio of partitios bega i the 20th cetury with the wors of S. Ramauja (97) ad G. H. Hardy. I particular G. H. Hardy (920) itroduced the otatios p ad q to represet the two most commoly used types of parititios. Defiitios of partitios The partitio fuctios discussed here iclude two basic fuctios that describe the structure of iteger umbers the umber of urestricted partitios of a iteger (partitios P) p, ad the umber of partitios of a iteger ito distict parts (partitios Q) q. Partitios P For oegative iteger, the fuctio p is the umber of urestricted partitios of the positive iteger ito a sum of strictly positive umbers that add up to idepedet of the order, whe repetitios are allowed. The fuctio p ca be described by the followig formulas: p t ; t p 0 ; 0, where t f t (with f t ) is the coefficiet of the t t term i the series expasio aroud t 0 of the fuctio f t : f t 0 t f t t. Example: There are seve possible ways to express 5 as a sum of oegative itegers: For this reaso p 5 7. Partitios Q For oegative iteger, the fuctio q is the umber of restricted partitios of the positive iteger ito a sum of distict positive umbers that add up to whe order does ot matter ad repetitios are ot allowed. The fuctio q ca be described by the followig formulas: q t t ;

2 2 q 0 ; 0, where t f t (with f t t ) is the coefficiet of the t term i the series expasio aroud t 0 of the fuctio f t : f t 0 t f t t. Example: There are three possible ways to express 5 as a sum of oegative itegers without repetitios: For this reaso q 5. Coectios withi the group of the partitios ad with other fuctio groups Represetatios through related fuctios The partitio fuctios p ad q are coected by the followig formula: 2 p q 2 p. 0 The best-ow properties ad formulas of partitios Simple values at zero ad ifiity The partitio fuctios p ad q are defied for zero ad ifiity values of argumet by the followig rules: p 0 q 0 p q. Specific values for specialized variables The followig table represets the values of the partitios p ad q for 0 0 ad some powers of 0: p q Aalyticity The partitio fuctios p ad q are o-aalytical fuctios that are defied oly for itegers. Periodicity The partitio fuctios p ad q do ot have periodicity. Parity ad symmetry The partitio fuctios p ad q do ot have symmetry. Series represetatios

3 The partitio fuctios p ad q have the followig series represetatios: p Π 2 A, sih Π Π2 p 9 A, 0F ; ; Π q 2 A 2, J 0 Π 2 24 q Π A 2, Π F ; 2;, where A, is a special case of a geeralized Kloosterma sum: A, h gcd h,, exp Π j j h j h j 2 2 Π h. Asymptotic series expasios The partitio fuctios p ad q have the followig asymptotic series expasios: p 4 exp 2 Π O ; q exp Π O ;. Geeratig fuctios The partitio fuctios p ad q ca be represeted as the coefficiets of their geeratig fuctios: p t ; t p t t 2 2 ; p t 8 2 t ; ϑ 0, t

4 q Σ q q q t t ; t ;, t 2 where t f t is the coefficiet of the t term i the series expasio aroud t 0 of the fuctio f t, f t 0 t f t t. Idetities The partitio fuctios p ad q satisfy umerous idetities, for example: p Σ p p p 2 2 p 2 2 p 2 p p 2 2 p 2 2 p 4 2 p Σ q 2 ;. Complex characteristics As real valued fuctios, the partitios p ad q have the followig complex characteristics: p q Abs p p q q Arg Arg p 0 Arg q 0 Re Re p p Re q q Im Im p 0 Im q 0 Cojugate p p q q Sig sg p sg q Summatio There exist just a few formulas icludig fiite ad ifiite summatio of partitios, for example: p 0 2 p t. t 0

5 5 Iequalities The partitios p ad q satisfy various iequalities, for example: p p p ; 2 q 2 q q ;. Cogruece properties The p partitios have the followig cogruece properties: p 5 4 mod 5 0 p 7 5 mod 7 0 p 6 mod 0 p mod ; 24 mod Zeros The p ad q partitios have the followig uique zeros: p 0 ; 0 q 0 ; 0. Applicatios of partitios Partitios are used i umber theory ad other fields of mathematics.

6 6 Copyright This documet was dowloaded from fuctios.wolfram.com, a comprehesive olie compedium of formulas ivolvig the special fuctios of mathematics. For a ey 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.