List of conjectural series for powers of $\pi$ and other constants
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Abstract:
Here I give the full list of my conjectures on series for powers of � and other important constants scattered in some of my public papers or my private diaries. The list contains 234 reasonable conjectural series. On the list there are 178 reasonable series for � 1 , four series for � 2 , two series for � 2 , four series for � 4 , two series for � 5 , three series for � 6 , seven series for �(3), one series for ��(3), two series for � 2 �(3), one series for �(3) 2 , three series involving both �(3) 2 and � 6 , one series for �(5), three series involving both �(5) and �(2)�(3), two series involving both ��(5) and � 3 �(3), three series involving �(7), three series for K = L(2,( · 3 )), one series for the Catalan constant G, two series forG, one series involving both � 3 G and � 2 �(3), two series forK, two series involving L = L(4,( ·Keywords:
Alternating series
Function series
Neumann series
Geometric series
General Dirichlet series
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Legendre function
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When we pause to reflect on Ramanujan’s life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathematics be brought to posterity. One of these was V. Ramaswamy Aiyer’s founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the next necessary episode, namely, Ramanujan’s meeting with Ramaswamy Aiyer at his office in Tirtukkoilur in 1910, would also have not taken place. Ramanujan had carried with him one of his notebooks, and Ramaswamy Aiyer not only recognized the creative spirit that produced its contents, but he also had the wisdom to contact others, such as R. Ramachandra Rao, in order to bring Ramanujan’s mathematics to others for appreciation and support. The large mathematical community that has thrived on Ramanujan’s discoveries for nearly a century owes a huge debt to V. Ramaswamy Aiyer. 1. THE BEGINNING. Toward the end of the first paper [57], [58 ,p . 36] that Ramanujan published in England, at the beginning of Section 13, he writes, “I shall conclude this paper by giving a few series for 1/π.” (In fact, Ramanujan concluded his paper a couple of pages later with another topic: formulas and approximations for the perimeter of an ellipse.) After sketching his ideas, which we examine in detail in Sections 3 and 9, Ramanujan records three series representations for 1/π .A s is customary, set
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Hypergeometric distribution
Basic hypergeometric series
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Hypergeometric identity
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Lauricella hypergeometric series
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We prove a Ramanujan-type formula for $520/\pi$ conjectured by Sun. Our proof begins with a hypergeometric representation of the relevant double series, which relies on a recent generating function for Legendre polynomials by Wan and Zudilin. After showing that appropriate modular parameters can be introduced, we then apply standard techniques, going back to Ramanujan, for establishing series for $1/\pi$.
Legendre function
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Congruence relation
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