In Praise of an Elementary Identity of Euler

نویسنده

  • Gaurav Bhatnagar
چکیده

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–Saalschütz 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 8φ7 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.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

The Process of Training Teachers for Elementary School in the Islamic Republic of Iran: Opportunities and Threats

The Process of Training Teachers for Elementary School in the Islamic Republic of Iran: Opportunities and Threats B. Sami’eenejaad M. Aliasgari, Ph.D. N. Moosaapoor, Ph.D. Gh. Haajihoseynnejaad, Ph.D. Teacher training in the Islamic Republic of Iran lacks a model matching its status. In order to shed light on all four aspects of training teachers for elementary school, ...

متن کامل

Representation of Racial Identity in Elementary and High Schools History Textbooks during the First Pahlavi Era

From the constructivism point of view, identity as a meaningful factor is not a constant matter but is current and variable. Essential components of identity also change and transform over time. Education is one of the most important institutions in which governments tend to inspire their intended identity towards society. The most important nationalistic components of Pahlavi government were b...

متن کامل

The So-called Renormalization Group Method Applied to the Specific Prime Numbers Logarithmic Decrease

A so-called Renormalization Group (RG) analysis is performed in order to shed some light on why the density of prime numbers in N decreases like the single power of the inverse neperian logarithm. CERN–TH/2000–146 May 2000 Part I The most elementary proof of the Prime Numbers Theorem These few lines are not part of the proof. They simply show the history which has led to the starting point of o...

متن کامل

Explanation and Prioritization of Elements of the Curriculum for Cultivation of Experimental Science in Elementary Education

Introduction: Creativity and innovation are one of the most advanced and complex activities of the human mind to be considered in education. The present study has been conducted with the aim of explaining and prioritizing elements of the curriculum of experimental science based on creativity development in the first elementary period. Methods: The present study, with a quantitative approach and...

متن کامل

Romantic Pedigree of Pahlavi the First’ Nationalism Discourse and National Identity Education in Primary Textbooks

As a modern autocracy, Reza Shah's government adopted a romantic authoritarian nationalist discourse that was rooted in the ideas of German thinkers such as Fichte and Herder as its paradigm for establishing Iranian national identity. This discourse insisted on the "national characteristics" and imagined the nation based on "romanticism" as an eternal phenomenon and national history was conside...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:
  • Electr. J. Comb.

دوره 18  شماره 

صفحات  -

تاریخ انتشار 2011