q-HYPERGEOMETRIC PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE’S IDENTITY AND EULER’S PENTAGONAL NUMBER THEOREM
نویسنده
چکیده
X iv :m at h/ 02 03 22 9v 1 [ m at h. C O ] 2 2 M ar 2 00 2 2000]Primary 05A19, 33D15 q-HYPERGEOMETRIC PROOFS OF POLYNOMIAL ANALOGUES OF THE TRIPLE PRODUCT IDENTITY, LEBESGUE’S IDENTITY AND EULER’S PENTAGONAL NUMBER THEOREM S. OLE WARNAAR Abstract. We present alternative, q-hypergeometric proofs of some polynomial analogues of classical q-series identities recently discovered by Alladi and Berkovich, and Berkovich and Garvan.
منابع مشابه
In Praise of an Elementary Identity of Euler
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 Ch...
متن کاملFranklin's Argument Proves an Identity of Zagier
Recently Zagier proved a remarkable q-series identity. We show that this identity can also be proved by modifying Franklin’s classical proof of Euler’s pentagonal number theorem. Mathematics Subject Classification (2000): 05A17 11P81
متن کاملEuler’s Pentagonal Number Theorem and the Rogers-fine Identity
Euler discovered the pentagonal number theorem in 1740 but was not able to prove it until 1750. He sent the proof to Goldbach and published it in a paper that finally appeared in 1760. Moreover, Euler formulated another proof of the pentagonal number in his notebooks theorem around 1750. Euler did not publish this proof or communicate it to his correspondents, probably because of the difficulty...
متن کاملSome Observations on Dyson's New Symmetries of Partitions
We utilize Dyson’s concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler’s Pentagonal Number Theorem. We streamline Dyson’s bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson’s adjoint of a partition to MacMahon’s “modular” partitions with modulus 2. This way we find a new combinato...
متن کاملBilateral truncated Jacobi's identity
Recently, Andrews and Merca considered the truncated version of Euler’s pentagonal number theorem and obtained a non-negative result on the coefficients of this truncated series. Guo and Zeng showed the coefficients of two truncated Gauss’ identities are non-negative and they conjectured that the truncated Jacobi’s identity also has non-negative coefficients. Mao provided a proof of this conjec...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006