A further look at the truncated pentagonal number theorem
نویسندگان
چکیده
منابع مشابه
The truncated pentagonal number theorem
A new expansion is given for partial sums of Euler’s pentagonal number series. As a corollary we derive an infinite family of inequalities for the partition function, p(n).
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In 2012 Andrews and Merca gave a new expansion for partial sums of Euler’s pentagonal number series and expressed k−1 ∑ j=0 (−1)(p(n− j(3j + 1)/2)− p(n− j(3j + 5)/2− 1)) = (−1)Mk(n) where Mk(n) is the number of partitions of n where k is the least integer that does not occur as a part and there are more parts greater than k than there are less than k. We will show that Mk(n) = Ck(n) where Ck(n)...
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Abstract. I revisit an automated proof of Andrews’ pentagonal number theorem found by Riese. I uncover a simple polynomial identity hidden behind his proof. I explain how to use this identity to prove Andrews’ result along with a variety of new formulas of similar type. I reveal an interesting relation between the tri-pentagonal theorem and items (19), (20), (94), (98) on the celebrated Slater ...
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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...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2019
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa180718-6-11