tailieunhanh - Báo cáo toán học: "n Praise of an Elementary Identity of Euler"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: In Praise of an Elementary Identity of Euler. | In Praise of an Elementary Identity of Euler Gaurav Bhatnagar Educomp Solutions Ltd. bhatnagarg@ Submitted Apr 14 2011 Accepted Jun 6 2011 Published Jun 11 2011 MSC2010 Primary 33D15 Secondary 11B39 33C20 33F10 33D65 Dedicated to S. B. Ekhad and D. Zeilberger Abstract We survey the applications of an elementary identity used by Euler in one of his proofs of the Pentagonal Number Theorem. Using a suitably reformulated version of this identity that we call Euler s Telescoping Lemma we give alternate proofs of all the key summation theorems for terminating Hypergeometric Series and Basic Hypergeometric Series including the terminating Binomial Theorem the Chu-Vandermonde sum the Pfaff-Saalschutz sum and their q-analogues. We also give a proof of Jackson s q-analog of Dougall s sum the sum of a terminating balanced very-well-poised 8O7 sum. Our proofs are conceptually the same as those obtained by the WZ method but done without using a computer. We survey identities for Generalized Hypergeometric Series given by Macdonald and prove several identities for q-analogs of Fibonacci numbers and polynomials and Pell numbers that have appeared in combinatorial contexts. Some of these identities appear to be new. Keywords Telescoping Fibonacci Numbers Pell Numbers Derangements Hypergeometric Series Fibonacci Polynomials q-Fibonacci Numbers q-Pell numbers Basic Hypergeometric Series q-series Binomial Theorem q-Binomial Theorem Chu-Vandermonde sum q-Chu-Vandermonde sum Pfaff-Saalschuutz sum q-Pfaff-Saalschutz sum q-Dougall summation very-well-poised 6 5 sum Generalized Hypergeometric Series WZ Method. 1 Introduction One of the first results in q-series is Euler s 1740 expansion of the product 1 - q 1 - q2 1 - q3 1 - q4 THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2 2011 P13 1 into a power series in q. This expansion known as Euler s Pentagonal Number Theorem is for q 1 1 - q 1 - q2 1 - q3 1 - q - q2 q5 q7 ----- E -. k k 3k 1 fc 3fc l -1 k q q . k 1 In his proof explained .