#A46 INTEGERS 18 (2018) JACOBI-TYPE CONTINUED FRACTIONS AND CONGRUENCES FOR BINOMIAL COEFFICIENTS

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1 #A46 INTEGERS 8 (08) JACOBI-TYPE CONTINUED FRACTIONS AND CONGRUENCES FOR BINOMIAL COEFFICIENTS Maie D. Schmidt School of Mathematics, Georgia Istitute of Techology, Atlata, Georgia maieds@gmail.com ; mschmidt34@gatech.edu Received: //7, Revised: /0/8, Accepted: 5//8, Published: 5/8/8 Abstract We prove two ew forms of Jacobi-type J-fractio epasios geeratig the biomial coe ciets, ad, over all 0. Withi the article we establish ew forms of iteger cogrueces for these biomial coe ciet variatios modulo ay (prime or composite) h ad compare our results with kow idetities for the biomial coe ciets modulo primes p ad prime powers p k. We also prove ew eact formulas for these biomial coe ciet cases from the epasios of the h th coverget fuctios to the ifiite J-fractio series geeratig these coe ciets for all.. Itroductio.. Cogrueces for Biomial Coe ciets Modulo Primes ad Prime Powers There are may well-kow results providig cogrueces for the biomial coe - ciets modulo primes ad prime powers. For eample, we ca state Lucas s theorem i the followig form for p prime ad, m N where = 0 p d p d ad m = m 0 m p m d p d for 0 apple i, m i < p [3]: 0 d (mod p). (Lucas Theorem) m m 0 m m d We also have kow results providig decompositios, or reductios of order, of the et biomial coe ciets modulo the prime powers, p k [5]: p k 0 p 0 mp k (mod p k ). m 0 mp m 0 Similarly, we ca decompose liear ad quadratic (ad more geeral) polyomial iputs to the lower biomial coe ciet idices as the cogruece factors give by

2 INTEGERS: 8 (08) [5, 3.]: p p (r ) rp s r s p p mp (m ) rp s m rp s (mod p ), (mod p 3 ). Withi the cotet of this article, we are motivated to establish meaigful, otrivial iteger cogrueces for the two idetermiate biomial coe ciet sequece variats, ad, modulo both prime ad composite bases h by usig ew epasios of the Jacobi-type cotiued fractios geeratig these biomial coe ciet variats that we prove i Sectio ad i Sectio 3. The et subsectio establishes related kow cotiued fractios ad iteger cogruece properties for the geeral forms of Jacobi-type J-fractios of which our ew results are special cases... Jacobi-Type Cotiued Fractio Epasios... Cotiued Fractios Geeratig the Biomial Coe ciets I this article, we study ew properties ad cogruece relatios satisfied by the iteger-order biomial coe ciets through two specific, ad apparetly ew, Jacobitype cotiued fractio epasios of a formal power series i z. The epasios of these J-fractios are similar i form to a kow Stieltjes-type cotiued fractio, or S-fractio, epasio give i Wall s book as [0, p. 343] ( z) k = kz (k) z ( k) 3 z (k) 3 4 z. (Biomial Series S-Fractio)... Jacobi-Type J-Fractios for Geeralized Factorial Fuctios The loosely-termed o-epoetial forms of a geeralized class of biomialcoe ciet-related geeralized factorial fuctios, p (, R), defied as, p (, R) := R(R )(R ) (R ( ) ), where the special cases of = p (, ), ( ) ( ) /, ad p (, R)/ ( ) R/ whe () = () ( ) deotes the Pochhammer symbol, are studied i [9]. Epoetial geeratig fuctios for the geeralized symbolic

3 INTEGERS: 8 (08) 3 product fuctios, p (, R), with respect to, are kow ad give i closed-form by X 0 p (, R ) z = ( z) (R)/, where the correspodig geeralized forms of the Stirlig umbers of the first kid, or -factorial coe ciets, are defied by the geeratig fuctios [8, cf. -3] X [R m ]p (, R ) z = ( )m m ( z) Log( z) m, for m Z. m 0 The power series epasios of the ordiary geeratig fuctios for these factorial fuctios, which typically coverge oly for z = 0 whe the parameter R is a fuctio of, are defied formally i [9] by ifiite J-fractios of the form Cov (, R; z) = R z (R ) z R z (R 4 ) z (R ) z 3 (R ) z These typically diverget ordiary geeratig fuctios are usually oly epaded as power series i z by regularized Borel sums ivolvig reciprocal powers of z (ad z) such as those give as eamples i the itroductio to [9]. This referece proves ew idetities for the geeralized factorial fuctios p (, R) by cosiderig the coe ciets of the ratioal h th covergets to the diverget power series defied oly formally by the previous equatio which are h order accurate i approimatig these geeratig fuctios. We employ a similar approach to eumeratig ew eact idetities ad cogruece properties of the biomial coe ciets i this article based o a formal treatmet of the ratioal covergets to the geeratig fuctios defied by two special cases of the ifiite Jacobi-type J-fractios defied i the et subsectio...3. Properties of the Epasios of Geeral J-Fractio Series More geerally, Jacobi-type J-fractios correspod to power series defied by ifiite cotiued fractio epasios of the form J [] (z) = c z c z ab z ab z,. (J-Fractio Epasios) for some sequeces {c i } i ad {ab i } i, ad some (typically formal) series variable z C [6, cf. 3.0] [0,, ]. The formal series eumerated by special cases of the

4 INTEGERS: 8 (08) 4 trucated ad ifiite cotiued fractio series of this form iclude ordiary (as opposed to typically closed-form epoetial) geeratig fuctios for may oe ad two-ide combiatorial sequeces icludig the geeralized factorial fuctios studied i [,, 4, 9]. The h th covergets, Cov h (z), to the J-fractio epasios of the form i the previous equatio have several useful characteristic properties: (A) The h th coverget fuctios eactly eumerate the coe ciets of the ifiite cotiued fractio series as [z ]J [] (z) = [z ] Cov h (z) for all 0 apple < h. (B) The coverget fuctio compoet umerator ad deomiator sequeces are each easily defied recursively by the same secod-order recurrece i h with di erig iitial coditios. The h th coverget fuctios are always ratioal i z which implies a evetually periodic ature to their coe ciets, as well as h-order fiite di erece equatios satisfied by the coe ciets of these trucated series approimatios. (C) If M h := ab ab h deotes the h th modulus of the geeralized J-fractio epasio defied above, the sequece of coe ciets eumerated by the h th coverget fuctio is evetually periodic modulo M h ad satisfies a fiite di erece equatio of order at most h [, Thm. ]. Moreover, if M m is itegervalued ad h divides M m for some m h, the [z ]J [] [z ] Cov m (z) (mod h) [4, 5.7]. Typically we are oly iterested i the cogrueces formed by the h th coverget fuctios for the coe ciets of z the ifiite series, J [] (z) i z for 0 apple apple h, which are i fact eactly eumerated by these coverget fuctios. Other properties of the coverget fuctios, such as fiite di erece equatios satisfied by their coe ciets, suggest other o-obvious ad o-trivial properties of the resultig cogrueces guarateed by the coverget fuctios..3. Orgaizatio of the Article We provide two ew forms of ifiite J-fractios geeratig the biomial coe ciets, ad, for 0 ad ay o-zero idetermiate i the et sectios of the article. These ew cotiued fractio results lead to ew eact formulas ad fiite di erece equatios for these biomial coe ciet variats, ad ew cogrueces for the biomial coe ciets modulo ay (prime or composite) itegers h wheever h h h h 3 / Z. h h Additio formulas for these J-fractios imply ew idetities for reductios of order of the upper ide,, comparable to the statemets of Lucas theorem ad the other results modulo primes p ad prime powers p k give i Sectio..

5 INTEGERS: 8 (08) 5 The proofs of these ew cotiued fractio epasios mostly follow by iductive argumets applied to kow recurrece relatios for the h th umerator ad deomiator coverget fuctio sequeces. The proofs that these ifiite J-fractio epasios eactly eumerate our biomial coe ciet variats of iterest follow from the ratioality of the h th coverget fuctios i z for all h. Cosequeces of the proofs of our mai theorems iclude ew eact formulas for the biomial coe - ciets, ad, ad ew cogruece properties for these biomial coe ciet variats. Sice the proofs of the correspodig results i each case give i Sectio ad Sectio 3, respectively, are so similar, we give careful proofs of the results for the case of the J-fractios geeratig state the aalogous results for the case of of epositio. h P,h(, z) z 3 ( )z ( )( )z ( 3)z ( 4)z 5 ( 3)( )z ( 3)( )( )z3 0 ( 4)( 3)z ( 4)( 3)( 6 )z3 ( 5)z ( 5)( 4)z 99 ( 5)( 4)( 3)z3 ( 5)( 4)( 3)( )( ) z 5 i the first sectio, ad choose to oly i the secod sectio below for clarity ( 4)( 3)( )( ) z ( 5)( 4)( 3)( ) z h (h ) P,h(, z) ( )z ( )z 6( )( )z ( 3)z 360( 3)( )z 4( 3)( )( )z ( 4)z 3040( 4)( 3)z 880( 4)( 3)( )z 3 0( 4)( 3)( )( )z ( 5)z ( 5)( 4)z 40300( 5)( 4)( 3)z 3 500( 5)( 4)( 3)( )z 4 70( 5)( 4)( 3)( )( )z 5 Table : The Numerator Coverget Fuctios, P,h (, z), Geeratig the Biomial Coe ciets,

6 INTEGERS: 8 (08) 6 h Q,h (, z) ( )z ( )z ( )( )z ( 3)z 3 ( )( 5 0 3)z ( )( )( 3)z ( 4)z ( 3)( 7 7 4)z ( )( 3)( 4)z ( )( )( 3)( 4)z4 5 5 ( 5)z 5 ( 4)( )z 5 ( 3)( 4)( 5 5)z3 5()(3)(4)(5) z ()()(3)(4)(5) z ( 6)z 3 ( 5)( 6)z ( 4)( 5)( 6)z3 99 ( 3)( 4)( 5)( 58 6)z4 ()(3)(4)(5)(6) z ()()(3)(4)(5)(6) z h (h ) Q,h (, z) ( )z 6 4( )z ( )( )z 3 0 7( 3)z 8( )( 3)z ( )( )( 3)z ( 4)z 70( 3)( 4)z 96( )( 3)( 4)z 3 6( )( )( 3)( 4)z ( 5)z 50400( 4)( 5)z 700( 3)( 4)( 5)z 3 600( )( 3)( 4)( 5)z 4 4( )( )( 3)( 4)( 5)z ( 6)z ( 5)( 6)z ( 4)( 5)( 6)z ( 3)( 4)( 5)( 6)z 4 430( )( 3)( 4)( 5)( 6)z 5 0( )( )( 3)( 4)( 5)( 6)z 6 Table : The Deomiator Coverget Fuctios, Q,h (, z), Geeratig the Biomial Coe ciets,. J-Fractio Epasios for the Biomial Coe ciets, Defiitio (Compoet Sequeces ad Coverget Fuctios for ). For a o-zero idetermiate, let the sequeces, c (),i ad ab(),i, be defied over i as follows: c (),i := ab (),i := 8 >< ( (i )i ), (i )(i 3) 4(i 3) ( i )( i ) if i 3 ( ) if i = >: 0 otherwise.

7 INTEGERS: 8 (08) 7 We the defie the h th coverget fuctios, Cov,h (, z) := P,h (, z)/ Q,h (, z), through the compoet umerator ad deomiator fuctios give recursively by P,h (, z) = ( c (),h z) P,h (, z) ab (),h z P,h (, z) [h = ] () Q,h (, z) = ( c (),h z) Q,h (, z) ab (),h z Q,h (, z) ( c (),z) [h = ] [h = 0]. Listigs of the first several cases of the umerator ad deomiator coverget fuctios, P,h (, z) ad Q,h (, z), are give i Table ad Table, respectively. Propositio (Formulas for the Numerator Coverget Fuctios). For all h ad idetermiate, the umerator coverget fuctios, P,h (, z), have the followig formulas: P h (, z) = =0 h (h ) (h ) (h ) (h ) ( z) = =0 h h h ( z). Proof. The proof follows from () of Defiitio by iductio o h. The claimed formula holds for h = 0,, by the computatios give i Table. Suppose that h 3 ad that the two equivalet formulas stated i the propositio for P,k (, z) are correct for all k < h. I particular, the stated formula holds whe k = h, h. The by epadig () usig our hypothesis, we have that P,k (, z) = ( h(h ) )z (h )(h 3) =0 h h h 3 ( z) ( h)( h )z 4(h 3) 3 =0 h h 3 h 5 ( z), which the implies the followig epasios:

8 INTEGERS: 8 (08) 8 h h 3 P,k (, z) = ( z) =0 h ( h(h ) ) X h h h 3 ( z) (h )(h 3) = h ( h)( h ) X h h 3 h 5 4(h 3) ( z) = h h h 3 ( h(h ) ) = ( z) (h )(h 3) ( h)( h ) h 3 h 5 4(h 3) h 3 h 3 h 3 h h 3 ( z) h h h h h h h (h )(h )( h) (h )(h )( h) i= ( h(h ) ) (h 3)(h )( h) ( )(h )( h)( h) (h 3)(h )(h )( h)( h) Fially, we ca simplify the last equatio ito the form of h h h P,k (, z) = h h h = ( z) ( z). ( z). h h ( z) h h h h The et idetity is used to prove Theorem below. I particular, the coe of the powers of z o the right-had-sides of the stated formulas satisfy ciets h h h ( ) () X i h h (h i) = ( ) i, i i (h i) (h ) i=0 which is proved directly by summig the correspodig hypergeometric terms eactly with Mathematica. Propositio (Formulas for the Deomiator Coverget Fuctios). For all h 0 ad fied idetermiates, we have that the deomiator coverget

9 INTEGERS: 8 (08) 9 fuctios, Q,h (, z), satisfy the followig formula: Q h (, z) = h i=0 i h (h i) z i. (h i) (h ) Proof. The proof agai follows from () of Defiitio by iductio o h. The claimed formula holds whe h = 0,, by the computatios give i Table. Net, we suppose that h 3 ad that the formula stated i the propositio for Q,k (, z) is correct for all k < h. I particular, the stated formulas hold whe k = h, h. The by epadig () usig our iductive hypothesis, we have that h (h ) (h 3 i) Q,h (, z) = z i i (h i) (h 3) i=0 ( h(h ) ) h (h ) (h )(h 3) i (h i) i= ( h)( h ) h 4(h 3) i i= = h hz (h ) h h(h )z h h (h ) = i=0 i= h i h (h i) (h ) (h i) (h i) z i (h 3) (h 3 i) z i (h 5) (h i) z i (h )(h )(h i)( h i) (h ) h(h i)(h i)(h ) (h )i( h(h ) ) h(h 3)(h i)(h ) (h )(h )(i(i )( h) h(h 3)(h i)(h i)(h ) h h (h i) z i. i (h i) (h ) The epasios of the J-fractios ad, more geerally, for ay cotiued fractio whose covergets are defied by the ratio of terms defied recursively as i (), provide additioal recurrece relatios ad eact fiite sums for the h th coverget fuctios, Cov,h (, z), give by [6,.]: Cov,h (, z) = Cov,h (, z) ( )h ab, ab,3 ab,h z h Q,h (, z) Q,h (, z) = ( )z i= ( ) i i i i i 3 i Q,i (, z) Q,i (, z) z i.

10 INTEGERS: 8 (08) 0 Theorem (Mai Theorem I). For itegers h, we have that the h th coverget fuctios, Cov,h (, z), eactly geerate the biomial coe ciets for all 0 apple apple h as [z ] Cov,h (, z) =. Proof. Sice Cov,h (, z) is ratioal i z for all h, Propositio ad Propositio imply the followig fiite di erece equatios for the coe ciets of the coverget fuctios, Cov,h (, z), whe 0 [4,.3]: [z ] Cov,h (, z) = mi(,h) X i= h [z i ] Cov,h (, z) h i (h i) [z ] P,h (, z) [0 apple < h]. (h i) ( ) i (h ) We must show that [z ] Cov,h (, z) = i the separate cases where 0 apple < h ad whe = h. For the first case, we use iductio o to show our result. Sice [z 0 ]F (z)/g(z) = F (0)/G(0) for ay fuctios, F (z) ad G(z), with a power series epasio i z about zero, we have by the two propositios above that [z 0 ] Cov,h (, z) = 0 0 for all h. Net, we suppose that for h > ad all k <, [z k ] Cov,h (, z) = k k. Sice 0 apple i < for all i i the right-had-side sum of (3), we ca apply the iductive hypothesis, combied with the observatio i () from the proof of Propositio, to the recurrece relatio i the previous equatio to obtai that whe < h we have X h i h [z (h i) ] Cov,h (, z) = ( ) i i i (h i) (h ) i= X h i h (h i) ( ) i i i (h i) (h ) i=0 =. To prove the claim i the first special case of (3) where [z ] P,h (, z) 0, i.e., precisely whe = h, we use a alterate approach to evaluatig these sums by eactly summig with Mathematica as X h i h [z (h i) ] Cov,h (, z) = ( ) i i i (h i) (h ) i= = ( 3 F ( h,, ( h); h, ( ); ), ) where p F q (a,..., a p ; b,..., b q ; z) deotes the geeralized hypergeometric fuctio whose coe ciets whe epaded i powers of z are give by the et Pochhammer (3)

11 INTEGERS: 8 (08) symbol products [6, 6]: pf q (a,..., a p ; b,..., b q ; z) = X k 0 (a ) k (a p ) k k (b ) k (b q ) k z k. Whe h =, the terms cotributed by the geeralized hypergeometric fuctio i the previous equatio are zero-valued. Therefore whe = h, we have that [z ] Cov,h (, z) =. We ca actually prove a stroger statemet of Theorem which provides that [z ] Cov,h (, z) = for all 0 apple < h, though for the purposes of showig that our J-fractio epasios defied by Defiitio are correct, the proof of the theorem give above su ces to show the result. Oe cosequece of the theorem is that we have the followig fiite sums eactly geeratig,, for ay ad all 0, i.e., i the case of (3) where h = : = i= X i ( ) i [ = 0]. i i i i Corollary (Cogrueces for Modulo Itegers h ). For h, let h() := Q h i= ab,i, or equivaletly, let h() ( )h h h h / h 3 h. For all h, m h, 0 apple apple h, ad 0, wheever m() Z ad h m(), we have the followig cogrueces cogrueces for the biomial coe ciets modulo h: X m i m (m i) ( ) i i i (m i) (m ) i= m m m [0 apple < m] (mod h). Proof. The result is a immediate corollary of (3) from the proof of Theorem ad the J-fractio coe ciet cogruece properties cited i property (C) of Sectio. i the itroductio [] [4, cf. 5.7]. Remark 3 (Eact Cogrueces for the Biomial Coe ciets). We cojecture that i fact for all h, h > 0, ad < h, which typically correspods to the cases of the biomial coe ciets that we actually wish to evaluate modulo i.e., sice ( ) k = 0 for all k 0 ad where for all itegers the et hypergeometric sum is evaluated eactly with Mathematica as X ( ) k k ( k) = 0. k ( ) k=0

12 INTEGERS: 8 (08) h, that h i i i= i h (h i) (h i) ( ) i (mod h), (h ) (4) at a miimum for ay such that h evely divides h() as a iteger factor, though other choices of the idetermiate may correctly yield cogruece modulo h. Give this restrictio o the modulo each h, we the defie the correspodig ide sets ( h h 3 M h := Z : Z). h h h h The first few particular special cases of these restricted ide sets iclude ( ) M = : Z = { : 0, 3 mod 4} 4 ( )( )( ) M 3 = : Z 6 = { : 0,, 7, 0, 6, 9, 5, 6 mod 7} ( )( )( )( )( 3) M 4 = : Z 8800 = { : 0,,, 7,, 7,, 3, 4 mod 5} \ { : 0,,, 3, 4, 7, 8, 9, 30, 3 mod 3} ( )( )( )( )( )( 3)( 4) M 5 = : Z = { : 0,,, 3,,, 3, 4 mod 5} \ { : 0,,, 3, 0, 7, 4, 3, 38, 45, 46, 47, 48 mod 49}. Now usig these results, we ca epad these special case cogrueces for modulo, 3, 4, ad 5 as follows: ( ) ( )( ) (mod ), 3 6 3( 3) 5 3( )( 3) 0 3 ( )( )( 3) 60 3 for all M (mod 3), for all M 3

13 INTEGERS: 8 (08) 3 4( 4) 7 ( 3)( 4) 7 3 ( )( 3)( 4) 05 ( )( )( 3)( 4) ( 5) 5( 4)( 5) ( 3)( 4)( 5) ( )( 3)( 4)( 5) ( )( )( 3)( 4)( 5) (mod 4), for all M 4 (mod 5), for all M 5. We also otice that oce we kow the restricted sets M h for ay h, the cogrueces epaded above provide us with ew iformatio about the cetral biomial coe ciets, for M h \ N, modulo prime ad composite choices of h. Eample 4 (Divisors of the Cetral Biomial Coe ciets). Oe famous eample of a problem we ca approach usig the ew machiery proved i this article is stated i [7, 4]. The problem (ad its prize) due to Ro Graham is to determie whether there are ifiitely may such that is relatively prime to 05, or equivaletly, to fid the sequece of such that oe of the idividual prime factors divide the th cetral biomial coe ciet: 3, 5, 7 -. Two of the cogrueces eplicitly epaded i the previous equatios imply ew geeratig fuctios for two of the three cases we eed to cosider i Graham s problem. For ay iteger modulus h, we set B cet,h (z) := X (mod p) z. M p\n We ca easily compute (i.e., usig Mathematica) that the o-idetically-zero coe ciets of the et geeratig fuctios, deoted by B e cet,h (z), are cogruet to the respective coe ciets of B cet,h (z) defied i the last equatio. For h {, 3}, particular cocrete eamples of these geeratig fuctios are epaded as eb cet, (z) = ˆB(z) 3 ˆB(z) 5 3 ˆB(z) 3 3 ˆB(z) 5 eb cet,3 (z) = 5 z ˆB(z) 64 ˆB(z) ˆB(z) ˆB(z) 5 56 ˆB(z) 7,

14 INTEGERS: 8 (08) 4 i powers of the ordiary geeratig fuctio for the cetral biomial coe defied by ˆB(z) := X z = p. 0 4z ciets It is easy to see by iductio from (4) that for higher-order h > 5, a geeral formula for the epasios of the geeratig fuctios B e cet,h (z) ca be epressed as a sum of a ratioal fuctio of z of small degree less tha h plus a fiite sum of odd powers of ˆB(z) m for m [, h ] \ Z \ Z. We ca the obtai formulas for most of the mod h whe M h by Faà di Bruo s formula or directly by takig the Cauchy products correspodig to multiple covolutios of the cetral biomial geeratig fuctio defied above. Corollary (Reductio of Order of the Biomial Coe ciets). For itegers r, p such that p r 0, let the sequeces, k r,p (), be defied as p r p r k r,p () = ( ) p r. p r r r For all itegers p, q 0, we have a additio theorem for the biomial coe ciets give by mi(p,q) p q X ( ) pq = k 0,p ()k 0,q () p q i= i()k i,p ()k i,q (), where h() is defied as i Corollary. Proof. We ca prove the result usig the matri method from [,, p. 33] ad [0, ]. I particular, for itegers p r 0, let the fuctios, e k r,p (), deote the solutios to the matri equatio 3 e k0, () e k, () 0 0 e k0, () e k, () e k, () e k0,3 () e k,3 () e k,3 () e k3,3 () e k0,0 () c, 0 0 e = k0, () e k, () 0 0 ab, c, e k0, () e k, () e k, () ab,3 c,3 7 5, (5) where here we defie e k 0,p () := p p for all p 0. We claim that for all itegers p, r 0 with p r 0, the two fuctios, k r,p () ad e k r,p (), are equal. To prove

15 INTEGERS: 8 (08) 5 the claim, we otice that for fied p, equatio (5) implies recurrece relatios over r give by e k,p () = ab, ek0,p () e kr,p () = c, ek 0,p (), p ab,r ekr,p () e kr,p () c,r ek r,p (), p > r. The first recurrece relatio provides that for p e p p k,p = ( ) p ( ) p ( ) ( ) p p = ( )p (p ) ( )(p ) ( p ) (p )( )(p ) p p ( )p p =, (p ) p which is the same as the formula for k,p () stated above. We ca the use the secod recurrece relatio to complete the proof of the claim by iductio o r. Whe r = 0,, we have that the two formulas for k r,p () ad e k r,p () coicide. We suppose that the claim is true for some r, which by the previous recurrece relatio i tur implies that e kr,p() = 4 (r ) ( r)( r ) = ( ) p r p p r ( ) p r p p r p p r p p ( ) r p p r p r p r r r ( )p r ( (r )(r ) ) p (r )(r ) p r r p r 4(r ) r r (r )( r ) p p r p p (p )(p )(r ) (p r)(p r)(r )(r ) (p r)(p r)(r )(r ) 4(p r)(p r)(r )(r ) ( (r )(r ) )(p r)(r ) (p r)(r )(r ) = ( ) p r p r p r. p r r r Sice the two formulas are equivalet, we obtai the et form of the additio formula, or alterately, a formula providig a reductio of the order of the upper

16 INTEGERS: 8 (08) 6 ad lower idices to the biomial coe ciets,, give by the epasio i the refereces i the followig forms for all itegers p, q 0: ( ) pq p q = k 0,pk 0,q ab, k,pk,q ab, ab,3 k,pk,q p q = ( ) pq p p mi(p,q) q q X ( ) pq i() i= p p i q q i i i p i q i. i i Eample 5. We otice that this result provides eact fiite sum epasios of the biomial coe ciets,, for ay ad 0 which hold modulo ay itegers h (prime or composite), ad compare the reductio i the upper ad lower coe ciet idices to the cogruece result provided by Lucas theorem ad its related variats stated i the itroductio. For a cocrete eample, let p = q = i Corollary. The we obtai that 4 = ( ) ( ) ( ( )( )( ) ) ( ) 7 ( 4)( 3)( )( ) =. 4 Furthermore, if we let Z ad specialize = p = q := i the corollary above, we are able to obtai the ew sum 3 X i = ( ) i i i i i= 6 = ( ). Related idetities ad cogrueces are formulated similarly by takig each of, p, q to be iteger multiples of the positive iteger specified i the previous eample. 3. J-Fractio Epasios for the Biomial Coe ciets, We ca costruct (ad formally prove) similar coverget-based J-fractio costructios eumeratig the biomial coe ciets,, for a idetermiate ad 0. Sice the proofs for the propositios, theorem, ad corollaries i this sectio are almost idetical to those give i the case of the biomial coe ciet variats,, i Sectio, we omit the proofs of the et results stated below. We begi with

17 INTEGERS: 8 (08) 7 the defiitio of the compoet sequeces ad coverget fuctios correspodig to the J-fractios geeratig the biomial coe ciets,, i this case. Defiitio 6 (Compoet Sequeces ad Coverget Fuctios for ). For a fied o-zero idetermiate ad i, we defie the compoet sequeces,, as follows: c (),i ad ab(),i c (),i := ab (),i := 8 >< (i )(i 3) (i ) 4(i 3) ( i )( i ) if i 3 ( ) if i = >: 0 otherwise. For h 0, we defie the h th coverget fuctios, Cov,h (, z) := P,h (, z)/ Q,h (, z), recursively through the compoet fuctios i the forms of P,h (, z) = ( c (),h z) P,h (, z) ab (),h z P,h (, z) [h = ] (6) Q,h (, z) = ( c (),h z) Q,h (, z) ab (),h z Q,h (, z) ( c (),z) [h = ] [h = 0]. Listigs of the first several cases of the umerator ad deomiator coverget fuctios, P,h (, z) ad Q,h (, z), are give i Table 3 ad Table 4, respectively. h P,h (, z) z 3 5 ( 3)z 0 ( )( 3)z ( 4)z 4 ( 3)( 4)z 0 ( )( 3)( 4)z ( 5)z ( 4)( 5)z 6 ( 3)( 4)( 5)z3 ()(3)(4)(5) 304 z ( 6)z ( 5)( 6)z 99 ( 4)( 5)( 6)z3 (3)(4)(5)(6) 584 z 4 ()(3)(4)(5)(6) z 5 Table 3: The Numerator Coverget Fuctios, P,h (, z), Geeratig the Biomial Coe ciets,

18 INTEGERS: 8 (08) 8 h Q,h (, z) z 3 4 ( 3 )z ( 6 )z 3 5 )z 3 ( 0 )( )z ( 60 )( )z3 4 ( 3)( )z ( )( )( )z3 ( 3)( )( )z ( 4)z 5 5 ( 4)( 3)z ( 4)( 3)( )z ( 4)( 3)( )( ) ( 4)( 3)( )( ) z z ( 5)z 3 ( 5)( 4)z ( 5)( 4)( 3)z3 99 ( 5)( 4)( 3)( )( ) ( 5)( 4)( 3)( )z4 58 ( 5)( 4)( 3)( )( ) z z 5 Table 4: The Deomiator Coverget Fuctios, Q,h (, z), Geeratig the Biomial Coe ciets, Propositio 3 (Coverget Fuctio Formulas). For all h ad a fied idetermiate, the umerator ad deomiator coverget fuctios, P,h (, z) ad Q,h (, z), each satisfy the followig respective formulas: h (h ) (h ) P,h (, z) = z (h ) (h ) =0 h h h = z =0 h i h (h i) Q,h (, z) = ( z) i. i (h i) (h ) i=0 Theorem 7 (Mai Theorem II). For all itegers h ad 0 apple apple h, we have that the h th coverget fuctios, Cov,h (, z), eactly geerate the biomial coe ciets as [z ] Cov,h (, z) =. As i Sectio, we remark that oe cosequece of the secod mai theorem resultig from the propositio above is that we have a ew proof of the et eact formula geeratig the biomial coe ciets,, for ay ad all 0: = X i= i i ( ) i [ = 0]. i i i Corollary 3 (Cogrueces for Modulo Itegers h ). Let the hth modulus, h(), of the J-fractio defied by Defiitio 6 correspod to the defiitios

19 INTEGERS: 8 (08) 9 give i Corollary. For all h, m h, 0 apple apple h, ad 0, wheever m() Z ad h m(), we have the followig cogrueces cogrueces for the biomial coe ciets,, modulo h: X m i m (m i) ( ) i i i (m i) (m ) i= m m m [0 apple < m] (mod h). Remark 8 (Eact Cogrueces for Special Cases). We agai cojecture that for all h ad all, 0, we have cogrueces for the biomial coe ciets,, of the form h i h h i i i i ( ) i h h h i= (mod h), for all M h, where the sets M h are defied as i Remark 3 of the previous sectio. We ca the epad the first several special cases of these cogrueces usig this result as follows: ( ) 3 3( ) 5 4( 3) 7 5( 4) 9 ( ) ( )( 3) [ apple ] (mod ), for all M 6 6 3( )( ) ( )( ) ( 3)( 4)( 5) 3 [ apple ] (mod 3), for all M 3 60 ( )( 3) ( )( )( 3) ( )( )( 3) ( 4)( 5)( 6)( 7) 4 [ apple 3] (mod 4), for all M ( 3)( 4) 5( )( 3)( 4) ( )( )( 3)( 4) ( )( )( 3)( 4) 50 5 ( 5)( 6)( 7)( 8)( 9) 5 [ apple 4] (mod 5), for all M Corollary 4 (Reductio Formulas for ). Let the fuctios, k r,s() be defied for all s r 0 as r r r s k r,s () =. s r r r

20 INTEGERS: 8 (08) 0 We have a correspodig additio, or lower ide reductio, formula for the biomial coe ciets,, give by the followig equatios for itegers p, q 0: p q pq mi(p,q) X = k 0,p ()k 0,q () = p q mi(p,q) X i= i= ( ) i i p i i()k i,p ()k i,q () pi i i q i qi i i i i i i i i. We ca agai compare the statemets of the two summatio formulas i the corollary to the biomial coe ciet epasios i the statemet of Lucas theorem from the itroductio modulo ay prime p. 4. Coclusios We have established the forms of two ew Jacobi-type J-fractio epasios providig ordiary geeratig fuctios for the biomial coe ciets, ad, for ay ad all 0. The key igrediets to the proofs of these series epasios are the ratioality of the h th covergets, Cov i,h (, z), for all h ad closed-form formulas for the correspodig umerator ad deomiator fuctio sequeces, which are fiite-degree polyomials i z with deg z {P i,h (, z)} = h ad deg z {Q i,h (, z)} = h for all h whe i =,. The fiite di erece equatios implied by the ratioality of the h th coverget fuctios at each h lead to ew eact formulas for, ad ew cogrueces satisfied by, the two biomial coe ciet variats studied withi the article. We ote that ulike the J-fractio epasios eumeratig the geeralized factorial fuctios studied i [9], the h th modulus, h(), of these J-fractios defied i Corollary is ot strictly iteger-valued for all ad h, which complicates the formulatios of the ew cogruece properties for the biomial coe ciet variats cosidered i the article. We also compare the two forms of the additio, or reductio of ide, formulas for these coe ciets stated i Corollary ad i Corollary 4 to the forms of Lucas s theorem ad to the prime power cogrueces cited i the itroductio, which similarly reduce the upper ad lower idices to these sequeces with respect to powers of prime moduli. New research directios based o the results i this article iclude further study of the biomial coe ciet cogrueces give i Sectio ad Sectio 3 modulo composite itegers h 4. I particular, oe directio for future research based o these topics is to fid eact formulas for doubly-ideed sequeces of multipliers, em i,h,, such that em,h, [z ] Cov,h (, z) (mod h) ad em,h,[z ] Cov,h (, z) (mod h) for all itegers, 0. This applicatio is surely a o-trivial task worthy of further study ad cosideratio based o the results we prove here.

21 INTEGERS: 8 (08) Refereces [] P. Flajolet, Combiatorial aspects of cotiued fractios, Discrete Math. 3 (980), 5 6. [] P. Flajolet, O cogrueces ad cotiued fractios for some classical combiatorial quatities, Discrete Math. 4 (98), [3] A. Graville, Biomial coe ciets modulo prime powers, adrew/biomial/ (996). [4] S. K. Lado, Lectures o Geeratig Fuctios, America Mathematical Society, USA, 00. [5] R. Me strović, Lucas theorem: Its geeralizatios, etesios ad applicatios (878 04), (04). [6] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, ad C. W. Clark, NIST Hadbook of Mathematical Fuctios, Cambridge Uiversity Press, 00. [7] C. Pomerace, Divisors of the middle biomial coe ciet, Amer. Math. Mothly (04). [8] M. D. Schmidt, Geeralized j factorial fuctios, polyomials, ad applicatios, J. Iteger Seq. 3 (00). [9] M. D. Schmidt, Jacobi-type cotiued fractios for the ordiary geeratig fuctios of geeralized factorial fuctios, J. Iteger Seq. 0 (07). [0] H. S. Wall, Aalytic Theory of Cotiued Fractios, Chelsea Publishig Compay, 948.