Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler

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1 Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio of Eule s petagoal umbe theoem ad of a idetity of Gauss by itoducig a ew paamete.. Itoductio Let = ( ( + ( ( be a biomial coefficiet. A. Beovich ad F. G. Gava [] ad with aothe method S.O. Waaa [] have poved that L ( + L ( = =L fo all L. This is a fiite vesio of Eule s petagoal umbe theoem, because fo L it educes to the petagoal umbe theoem = ( + ( = (. = Let ( x, a = x a. A famous theoem of Gauss states that + (, = ad ( ; (, = = ( ( (. ( ; Geealizig these esults we give explicit evaluatios of the sums L ( + + L hl (, = ( =L ad (, = (. =

2 . Vaiatios of a fiite vesio of Eule s petagoal umbe theoem Geealizig the fomula of Beovich ad Gava we give explicit evaluatios of the sums L ( + + L hl (, = (. =L (. Fo small values of we get the followig fomulas fo hl (, : hl (, = L hl (, = L hl (, = + L L hl (, = + L L L hl (, = + Fo = the left had side of each euatio educes to. The ight had side educes to a diffeece of sums of the fom Ai, =. i(mod E.g A7, A7, = = ( (+ =. The existece of 7 5 such a epesetatio becomes obvious fom ( + ρ = A, + A, ρ+ A, ρ = ( ρ, whee ρ deotes a thid oot of uity. m+ m+ m E.g. if = m+ we have ( ρ = ( ρ = ( ( + ρ. Theefoe m A, A, = (. I the geeal case we defie a seuece w ( fo by w ( = if (mod ad ( 6 w ( = ( else. This meas w( = ( ( w(+ = ( w(+ =. (+ (.

3 Fom the defiitio follows that w( = w( +. (. The we have Theoem Fo each L ad the sum L ( + + L hl (, = ( =L has the value + + L L (. hl (, = ( w( ( w (. = + + = = Sice w( = if (mod thee emai oly biomial coefficiets L with (mod o + (mod. The sig of the coefficiet of is give by ( = (. Theefoe all tems with i the same esidue class mod have the same sig. hl (,7 = A ( L, A ( L, with E.g. 7, 7, 7 7 A L 5 L 5L 7,(, = +. L 7 L 7 7L7 7 A7,( L, = ad The expasio of hl (, has a vaishig costat tem if ad oly if we get h(+ = lim h( L, + =. L Fo = the costat tem is Aalogously we get h( = (. Fom the defiitio of hl (, we see that = (+ w( = ( ad theefoe ( ( + + ( + i h( + i ( = (. = (mod. Theefoe (+ h( = (. This ca also be veified by usig Jacobi s tiple poduct idetity. The case h ( = gives Eule s petagoal umbe theoem. Fo the tiple poduct implies ( a b + ( a + a + b ( a + a b ( a + = a. = Theefoe = ( + + ( + i i + i + ( = ( ( (.

4 Fo i = l+ the poduct vaishes because oe of the middle factos vaishes. Fo i = l we get ( + l(( + l i + i + ( ( ( = ( = h( + i ( + because to each tem, < + l, which does ot occu i the fist factos of the ( poduct thee coespods pecisely oe tem + i the middle factos ad ( + l(( + l (( + l =. The case i = l ca be teated i the same way. Poof of Theoem I ode to pove this theoem we use some esults about the Fiboacci polyomials + F ( x, s = x s = fom ou pevious pape []. These ae aalogues + f (x,s = x s, = of the Fiboacci polyomials which ae chaacteized by the ecuece elatio f ( x,s = xf ( x,s + sf ( x,s ad the iitial values f (x,s= ad f (x,s =. Let ad i ( i i GLis (,, = s =L (.5 + f( s = s. = (.6 The f( s = F(, s ad theefoe f ( s = f ( s + sf ( s (.7 by [] (..

5 Usig the easily veified fomula we get + = ( ( L L + i + i + il i + L i i s = s. (.8 = L = The (.8 ca be witte as L( + il L GLis (,, = s f ( s. (.9 + i Combiig (.7 ad (.9 we get L GLis (,, = GLi (,, s + sgli (,, s. (. By [] (. ( ( f ( =, f ( = (, f ( = (. (. (, The fomula (, ( F s F s = (cf. [] (.7 s gives F (, =F(,. Theefoe we get ( + ( + + F (, = (, F (, = (, F (, =. This implies that (. holds fo all. We use (.9 to exted GLis (,, to values with i+ <. The it is easy to veify that GL (,, = w ( (. fo all. Note that the ight-had side does ot deped o L. 5

6 Fomula (. gives immediately This follows by iductio fom L GLis (,, = s GLi (, s,. = (. GLi s GLi s sgli s + (,, = (,, + (,, L L = s G( L, i, s + s G( L, i, s = = L L = s G( L, i, s + s G( L, i, s + L + = s G( L, i, s. = The theoem follows if i (. we set s = ad i =.. Vaiatios of a -idetity of Gauss Coside ow the Roges-Szegö polyomials ( x, a = x a. (. Gauss s theoem states that + (, = ad = ( ; = (. (, ( ( (. ( ; A vey simple poof uses Eule s expoetial seies x ex ( = = =. ( x; ( x( x( x ( ; (. The geeatig fuctio of the Roges-Szegö polyomials is give by ( x, a z = e ( xz e ( az. (. ( ; This follows fom + xz az z ( ; + z + exzeaz ( ( = = xa = xa ( ; ( ;, ( ; ( ; ( ; ( ; + + = ( x, a z =. ( ; 6

7 As a special case we get (, z = e( z e( z = ( ; ( + z( + z( + z ( z( z( z ( z z = = = ( ; ( z ( z ( z ( ; ( ( (, (.5 which is euivalet with Gauss s theoem by compaig coefficiets. B.A Kupeshmidt [] has give a fomula fo (, x = ( ( x, which ca be easily deduced fom the geeatig fuctio ( x, exz ( z = e( xz e( z = ( e( z e( z. ( ; ez ( He aised the poblem of explicitly evaluatig (,. I the followig we give two such fomulas. Fo the fist oe we geealize the method we used to pove Gauss s theoem: Fom ex ( = ( xex ( ( x( x = we get (, + z z = e( z e( z = ( ; ( + z( + z( + z ( z( z( z + z ( z z = = ( +. ( z ( z ( z ( ; This implies (, = (, ad (, = (,. Moe geeally fo (, ( + z ( + z z = e( z e( z = ( ; ( + z( + z( + z ( z( z( z (, z ( ; = ( ; = ( + z ( + z z = z. Compaig coefficiets we get (, = ( ;. + = ( ; (.6 This may be witte i the fom ( ; (, = (, < ( ; + i = (, ( + i= 7

8 ad i. (, = (, ( i= + The fist values ae (, = (, (, = (, (, = + ( (, (, = + ( (, (, = + ( + ( ( (, 5 ad (, = (, (, (, (, = + ( = + ( (, (, 6 = + ( + ( (. (, These ae polyomials i. Now we wat to compute the coefficiets of these polyomials. I ode to do this we stat fom the fomula x + x ( x, a ( ( x, a ( x, a =, (.7 a a which is easily veified by compaig coefficiets: + + ( (. + = + = 8

9 Let ow = (,. b (, (, (.8 The we get + + b (, + ( + b (, + + b (, =. (.9 Defie ow polyomials f (,s by the ecuece f ( s, = ( + f(, s sf(, s (. ad iitial values f (,s= ad f (,s =. The b (, = f(,. (. The polyomial f (,sis a aalogue of the Fiboacci polyomial f (, s = ( s. = It is easily veified that this fomula has the diect aalogue f (,s = ( s ( +. i (. = i= Fo by compaig coefficiets the ecusio (. is euivalet with the idetity ( ( ( = This is tivial, because we get + = by usig both ecueces fo the biomial coefficiets. Fom + + (, + (, + (, = (. (, (, (, we see that (, c b b (, (, = = ( ( +, + ( +,. (. 9

10 This gives [ ] i [ = ]. i= c (, = ( ( + Theefoe we have poved Theoem Fo each fixed the followig idetities hold: (, = + (, i ( ( (. = i= ad [ ] i ( (. = [ ] (.5 i= (, = + (, Refeeces [] A. Beovich ad F.G. Gava, Some obsevatios o Dyso s ew symmeties of patitios, J. Comb. Th., Se. A (, 6-9 [] J. Cigle, A ew class of -Fiboacci polyomials. Elect. J. Comb. (, #R9 [] B.A. Kupeshmidt, -Newto Biomial: Fom Eule to Gauss, Joual of Noliea Math. Physics, V.7, N (, -6 [] S. O. Waaa, -hypegeometic poofs of polyomial aalogues of the tiple poduct idetity, Lebesgue s idetity ad Eule s petagoal umbe theoem, Ramaua J. 8 (, 67-7

SHIFTED HARMONIC SUMS OF ORDER TWO

Commu Koea Math Soc 9 0, No, pp 39 55 http://dxdoiog/03/ckms0939 SHIFTED HARMONIC SUMS OF ORDER TWO Athoy Sofo Abstact We develop a set of idetities fo Eule type sums I paticula we ivestigate poducts of