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Journal articles on the topic 'Quintuple product identity'

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Published: 4 June 2021

Last updated: 4 February 2022

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1

COOPER, SHAUN. "THE QUINTUPLE PRODUCT IDENTITY." International Journal of Number Theory 02, no. 01 (March 2006): 115–61. http://dx.doi.org/10.1142/s1793042106000401.

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The quintuple product identity was first discovered about 90 years ago. It has been published in many different forms, and at least 29 proofs have been given. We shall give a comprehensive survey of the work on the quintuple product identity, and a detailed analysis of the many proofs.

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2

Farkas, Hershel M., and Irwin Kra. "On the quintuple product identity." Proceedings of the American Mathematical Society 127, no. 3 (1999): 771–78. http://dx.doi.org/10.1090/s0002-9939-99-04791-7.

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3

Hirschhorn, M. D. "A generalisation of the quintuple product identity." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 44, no. 1 (February 1988): 42–45. http://dx.doi.org/10.1017/s1446788700031359.

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AbstractThe quintuple product identity has appeared many times in the literature. Indeed, no fewer than 12 proofs have been given. We establish a more general identity from which the quintuple product identity follows in two ways.

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4

Chan, Song Heng, Thi Phuong Nhi Ho, and Renrong Mao. "Truncated series from the quintuple product identity." Journal of Number Theory 169 (December 2016): 420–38. http://dx.doi.org/10.1016/j.jnt.2016.05.013.

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5

Chen, William Y. C., Wenchang Chu, and Nancy S. S. Gu. "Finite form of the quintuple product identity." Journal of Combinatorial Theory, Series A 113, no. 1 (January 2006): 185–87. http://dx.doi.org/10.1016/j.jcta.2005.04.002.

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6

KIM, SUN. "A BIJECTIVE PROOF OF THE QUINTUPLE PRODUCT IDENTITY." International Journal of Number Theory 06, no. 02 (March 2010): 247–56. http://dx.doi.org/10.1142/s1793042110002909.

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7

Chan, Hei-Chi. "Another simple proof of the quintuple product identity." International Journal of Mathematics and Mathematical Sciences 2005, no. 15 (2005): 2511–15. http://dx.doi.org/10.1155/ijmms.2005.2511.

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8

Alladi, Krishnaswami. "The quintuple product identity and shifted partition functions." Journal of Computational and Applied Mathematics 68, no. 1-2 (April 1996): 3–13. http://dx.doi.org/10.1016/0377-0427(95)00251-0.

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9

CHEN, SIN-DA, and SEN-SHAN HUANG. "ON GENERAL SERIES-PRODUCT IDENTITIES." International Journal of Number Theory 05, no. 06 (September 2009): 1129–48. http://dx.doi.org/10.1142/s1793042109002596.

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We derive the general series-product identities from which we deduce several applications, including an identity of Gauss, the generalization of Winquist's identity by Carlitz and Subbarao, an identity for [Formula: see text], the quintuple product identity, and the octuple product identity.

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10

Zhu, Jun-Ming, and Zhi-Zheng Zhang. "A semi-finite form of the quintuple product identity." Journal of Combinatorial Theory, Series A 184 (November 2021): 105509. http://dx.doi.org/10.1016/j.jcta.2021.105509.

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11

Chan, Heng Huat. "Triple product identity, Quintuple product identity and Ramanujan's differential equations for the classical Eisenstein series." Proceedings of the American Mathematical Society 135, no. 07 (July 1, 2007): 1987–93. http://dx.doi.org/10.1090/s0002-9939-07-08723-0.

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12

Srivastava, Bhaskar. "A new form of the quintuple product identity and its application." Filomat 31, no. 7 (2017): 1869–73. http://dx.doi.org/10.2298/fil1707869s.

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We give a new form of the quintuple product identity. As a direct application of this new form a simple proof of known identities of Ramanujan and also new identities for other well known continued fractions are given. We also give and prove a general identity for (q3m; q3m)?.

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13

Bhargava, S., Chandrashekar Adiga, and M. S. Mahadeva Naika. "QUINTUPLE PRODUCT IDENTITY AS A SPECIAL CASE OF RAMANUJAN'S 1ψ1 SUMMATION FORMULA." Asian-European Journal of Mathematics 04, no. 01 (March 2011): 31–34. http://dx.doi.org/10.1142/s1793557111000046.

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In this note we observe an interesting fact that the well-known quintuple product identity can be regarded as a special case of the celebrated 1ψ1 summation formula of Ramanujan which is known to unify the Jacobi triple product identity and the q -binomial theorem.

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14

Liu, Zhi-Guo. "An extension of the quintuple product identity and its applications." Pacific Journal of Mathematics 246, no. 2 (June 1, 2010): 345–90. http://dx.doi.org/10.2140/pjm.2010.246.345.

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15

Bhargava, S., Chandrashekar Adiga, and M. S. Mahadeva Naika. "Ramanujan's remarkable summation formula as a $2$-papameter generalization of the quintuple product identity." Tamkang Journal of Mathematics 33, no. 3 (September 30, 2002): 285–88. http://dx.doi.org/10.5556/j.tkjm.33.2002.285-288.

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It is well known that `Ramanujan's remarkable summation formula' unifies and generalizes the $q$-binomial theorem and the triple product identity and has numerous applications. In this note we will demonstrate how, after a suitable transformation of the series side, it can be looked upon as a $2$-parameter generalization of the quintuple product identity also.

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16

Hammond, Paul, Richard Lewis, and Zhi-Guo Liu. "Hirschhorn's identities." Bulletin of the Australian Mathematical Society 60, no. 1 (August 1999): 73–80. http://dx.doi.org/10.1017/s0004972700033347.

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We prove a general identity between power series and use this identity to give proofs of a number of identities proposed by M.D. Hirschhorn. We also use the identity to give proofs of a well-known result of Jacobi, the quintuple-product identity and Winquist's identity.

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17

Clark, J. S., M. E. Lohr, L. R. Patrick, F. Najarro, H. Dong, and D. F. Figer. "An updated stellar census of the Quintuplet cluster." Astronomy & Astrophysics 618 (October 2018): A2. http://dx.doi.org/10.1051/0004-6361/201833041.

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Context. Found within the central molecular zone, the Quintuplet is one of the most massive young clusters in the Galaxy. As a consequence it offers the prospect of constraining stellar formation and evolution in extreme environments. However, current observations suggest that it comprises a remarkably diverse stellar population that is difficult to reconcile with an instantaneous formation event. Aims. To better understand the nature of the cluster our aim is to improve observational constraints on the constituent stars. Methods. In order to accomplish this goal we present Hubble Space Telescope/NICMOS+WFC3 photometry and Very Large Telescope/SINFONI+KMOS spectroscopy for ∼100 and 71 cluster members, respectively. Results. Spectroscopy of the cluster members reveals the Quintuplet to be far more homogeneous than previously expected. All supergiants are classified as either O7–8 Ia or O9–B0 Ia, with only one object of earlier (O5 I–III) spectral type. These stars form a smooth morphological sequence with a cohort of seven early-B hypergiants and six luminous blue variables and WN9-11h stars, which comprise the richest population of such stars of any stellar aggregate known. In parallel, we identify a smaller population of late-O hypergiants and spectroscopically similar WN8–9ha stars. No further H-free Wolf–Rayet (WR) stars are identified, leaving an unexpectedly extreme ratio of 13:1 for WC/WN stars. A subset of the O9–B0 supergiants are unexpectedly faint, suggesting they are both less massive and older than the greater cluster population. Finally, no main sequence objects were identifiable. Conclusions. Due to uncertainties over which extinction law to apply, it was not possible to quantitatively determine a cluster age via isochrone fitting. Nevertheless, we find an impressive coincidence between the properties of cluster members preceding the H-free WR phase and the evolutionary predictions for a single, non-rotating 60 M⊙ star; in turn this implies an age of ∼3.0–3.6 Myr for the Quintuplet. Neither the late O-hypergiants nor the low luminosity supergiants are predicted by such a path; we suggest that the former either result from rapid rotators or are the products of binary driven mass-stripping, while the latter may be interlopers. The H-free WRs must evolve from stars with an initial mass in excess of 60 M⊙ but it appears difficult to reconcile their observational properties with theoretical expectations. This is important since one would expect the most massive stars within the Quintuplet to be undergoing core-collapse/SNe at this time; since the WRs represent an evolutionary phase directly preceding this event,their physical properties are crucial to understanding both this process and the nature of the resultant relativistic remnant. As such, the Quintuplet provides unique observational constraints on the evolution and death of the most massive stars forming in the local, high metallicity Universe.

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18

Ma, X. "Two Finite Forms of Watson's Quintuple Product Identity and Matrix Inversion." Electronic Journal of Combinatorics 13, no. 1 (June 12, 2006). http://dx.doi.org/10.37236/1078.

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Recently, Chen-Chu-Gu and Guo-Zeng found independently that Watson's quintuple product identity follows surprisingly from two basic algebraic identities, called finite forms of Watson's quintuple product identity. The present paper shows that both identities are equivalent to two special cases of the $q$-Chu-Vandermonde formula by using the ($f,g$)-inversion.

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19

Chu, Wenchang, and Qinglun Yan. "Unification of the Quintuple and Septuple Product Identities." Electronic Journal of Combinatorics 14, no. 1 (March 28, 2007). http://dx.doi.org/10.37236/1008.

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By combining the functional equation method with Jacobi's triple product identity, we establish a general equation with five free parameters on the modified Jacobi theta function, which can be considered as the common generalization of the quintuple, sextuple and septuple product identities. Several known theta function formulae and new identities are consequently proved.

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20

Paule, Peter. "Short and Easy Computer Proofs of the Rogers-Ramanujan Identities and of Identities of Similar Type." Electronic Journal of Combinatorics 1, no. 1 (July 26, 1994). http://dx.doi.org/10.37236/1190.

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New short and easy computer proofs of finite versions of the Rogers-Ramanujan identities and of similar type are given. These include a very short proof of the first Rogers-Ramanujan identity that was missed by computers, and a new proof of the well-known quintuple product identity by creative telescoping.

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