ON DIFFERENTIATION AND HARMONIC NUMBERS
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1 ON DIFFERENTIATION AND HARMONIC NUMBERS ERIC MORTENSON Abstract. I a paper of Adrews ad Uchimura [AU, it is show how differetiatio applied to hypergeometric idetities produces formulas for harmoic ad q-harmoic umbers. Here we recall two biomial coefficiet sums that appear i [M, ad reprove them usig the techiques of [AU. 1. Itroductio ad Goal It is kow that may biomial coefficiet formulas follow from hypergeometric series idetities [A1. However, as is demostrated i [AU there are biomial coefficiet idetities which are ot ecessarily of this type. Here we recall two biomial coefficiet sums from [M ad reprove them usig the methods of [AU. We first recall idetities from (6.21 ad (5.28 i [M: (1.1 (1.2 ( 1 (H+ + H m+ 2H 0, ( 1 (1 + (H + + H m+ 2H ( 1 (1 + + m, where H : For both idetities we have the coditio m 1. I [M, these idetities were crucial i provig several Beukers like supercogrueces that had bee observed umerically by Ferado Rodriguez-Villegas [FRV. I [M, these idetities were broke up ito smaller pieces, ad each part was evaluated usig Wilf-Zeilberger [PWZ theory. Although these were evaluated exactly, for the goal of [M oe oly eeded to show that the two sums were cogruet to zero modulo p, where + m p 1 ad p 5 was prime. Our obective here is two-fold: to show that this ca be 2000 Mathematics Subect Classificatio. 05A19. The author thaks the Natioal Sciece Foudatio for their geerous support. 1
2 2 ERIC MORTENSON accomplished from differetiatio of classical hypergeometric idetities ad to brig attetio to this useful techique foud i [AU. We recall some basic defiitios, ( x + + (1.3 δ[f(x : f (0 ad where (1.4 (a : a(a + 1 (a + 1. Thus we ca write [ ( x + + ( + (1.5 δ (H + H, ad : (x (1, ( ( 1. We also recall the stadard otatio for hypergeometric series [A2, (1.6 2F 1 ( a, b c x 0 as well as the Chu-Vadermode sum [A2: (1.7 2F 1 (, a c 1 (a (b (c x, (c a (c. 2. The First Idetity Here we break the secod idetity up ito two parts. For the first part, we proceed as follows: ( 1 (H + H [ [ (x + 1 [ (x + 1 (x [ (x + 1 2F 1 (, x x ( (x + 1 [( 1 0 ( + (H + H (x (x + 1
3 ON DIFFERENTIATION AND HARMONIC NUMBERS 3 The secod part is aalogous, but we replace with p m 1, ad the cosider it mod p. ( 1 ( + 1 (H m+ H (H m+ H m (m p + 1 ( m (p m (H m+ H (m + 1 (H m+ H (mod p 0 Puttig both parts together, it follows that the first idetity is cogruet to zero mod p. 3. The Secod Idetity We break the first idetity up ito three sums, ad aalyze each separately. The first is well kow, but we iclude it for the sake of completeess. ( + ( 1 ( ( + 1! 2 F 1 (, For the secod sum, we proceed as follows: ( 1 ((H + H [ (x [ (x + 1 (x + + 1( (x + 1 [ (x ( ( 1. (1 1 ( + 1 ( 1 ( 1! (x ( 1! (x + 1 (x ( 1 (x + 2 ( 1 [ ( (x + 1 (x + + 1( + 1, x F 1 1 (x + 1 x + 2 [ (x + 1 (x + + 1( ( 1 (x + 1 (x [ (x + 1 (x + + 1( ( 1 1 (x + 1 [ (x + + 1( ( 1 1 ( 1.
4 4 ERIC MORTENSON for the third ad fial sum, we recall that +m p 1 ad argue as before ( 1 ((H m+ H [ (x + 1m δ m! m 1 ( 1 m m ( 1 m ( m (x + m + 1 ( 1! (x + 1 (mod p The last equality follows because + m p 1 is eve, thus ad m have the same parity. Therefore the sum is cogruet mod p to ( 1 (1 + + m ( 1 p, i.e. zero mod p. ( Remark The referee speculated as to the truth of the followig geeral idetity ( ( x + m ( 1 ( k + 1 k1 ( (x + 1 m, m which would shorte the proofs i the previous two sectios. Ideed, replace x with x + ad the let let m 0, 1 i sectios 2 ad 3 respectively. Provig this idetity is a opportuity to emphasize this ote s uderlyig theme: write it as a hypergeometric series ad the use Chu-Vadermode to trasform it. Replacig x with x + for coveiece, we have ( ( x + + m ( 1 ( k + 1 m (x + 1 (x + 1 (x + 1 k1 (x (x + 1 ( m! (x m ( 1 ( m (x m ( m (x + 1 m m (x m ( m (x + 1 m 2F 1 m (x m ( m ( m (x + 1 m (x + m + 1 m ( 1 m ( 1 m ( m! m! (x m. (x ( m! (x + 1 ( + m ( m (x + + m + 1 ( m ( m! (x + m + 1 ( m ( + m, x + + m x + m + 1
5 ON DIFFERENTIATION AND HARMONIC NUMBERS 5 5. Ackowledgemets The author would like to thak the referee for the helpful commets ad suggestios which improved this paper. Refereces [A1 G. Adrews, Applicatios of Hypergeometric Fuctios, S.I.A.M. Review, 16 (1974, pages [A2 G. Adrews, The Theory of Partitios, Vol. 2, Ecycl. of Math. ad Its Appl., Addiso-Wesley, Readig, (reprited: Cambridge Uiversity Press, [AU G. Adrews, K. Uchimura, Idetities i Combiatorics IV: Differetio ad Harmoic Numbers, Utilitas Mathematica 28 (1985, pages [FRV F. Rodriguez-Villegas, Hypergeometric Families of Calabi-Yau Maifolds. Calabi-Yau varieties ad mirror symmerty, Fields Ist. Commu., 38, Amer. Math. Soc., Providece, RI, [M E. Morteso, Supercogrueces Betwee Trucated 2 F 1 Hypergeometric Fuctios ad their Gaussia Aalogs, Tras. Amer. Math. Soc., 335 (2003, pages [PWZ M. Petkovsek, H. Wilf, D. Zeilberger, AB, A. K. Peters, Ltd., Wellesley, Ma, Departmet of Mathematics, Pe State Uiversity, Uiversity Park, Pesylvaia address: mort@math.psu.edu
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