A new class of infinite products, and Euler's totient
We introduce some new infinite products, the simplest being(1−y)∏k=2∞∏j∈ϕk(1−ykqj)1/k=(1−y1−qy)1/(1−q),where ϕk is the set of positive integers less than and relatively prime to k, valid for |y|∧|qy| both less than unity, with q≠1. The idea of a q-analogue for the Euler totient function is suggested...
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| Format: | Article |
| Language: | English |
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Wiley
1994-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171294000591 |
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| _version_ | 1832551538900336640 |
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| author | Geoffrey B. Campbell |
| author_facet | Geoffrey B. Campbell |
| author_sort | Geoffrey B. Campbell |
| collection | DOAJ |
| description | We introduce some new infinite products, the simplest being(1−y)∏k=2∞∏j∈ϕk(1−ykqj)1/k=(1−y1−qy)1/(1−q),where ϕk is the set of positive integers less than and relatively prime to k, valid for |y|∧|qy| both less than unity, with q≠1. The idea of a q-analogue for the Euler totient function is suggested. |
| format | Article |
| id | doaj-art-6a6de85818b44b1085d47bfafc0e6631 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1994-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-6a6de85818b44b1085d47bfafc0e66312025-02-03T06:01:13ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251994-01-0117341742210.1155/S0161171294000591A new class of infinite products, and Euler's totientGeoffrey B. Campbell0Mathematics Research Section, Institute of Advanced Studies, School of Mathematical Sciences, The Australian National University, GPO Box 4, Canberra 2601, AustraliaWe introduce some new infinite products, the simplest being(1−y)∏k=2∞∏j∈ϕk(1−ykqj)1/k=(1−y1−qy)1/(1−q),where ϕk is the set of positive integers less than and relatively prime to k, valid for |y|∧|qy| both less than unity, with q≠1. The idea of a q-analogue for the Euler totient function is suggested.http://dx.doi.org/10.1155/S0161171294000591combinatorial identitiespartitionsarithmetic functionsconvergence and divergence of infinite productslattice points in large regionsapplications of sieve methodscombinatorial enumeration problemsgenerating functions. |
| spellingShingle | Geoffrey B. Campbell A new class of infinite products, and Euler's totient International Journal of Mathematics and Mathematical Sciences combinatorial identities partitions arithmetic functions convergence and divergence of infinite products lattice points in large regions applications of sieve methods combinatorial enumeration problems generating functions. |
| title | A new class of infinite products, and Euler's totient |
| title_full | A new class of infinite products, and Euler's totient |
| title_fullStr | A new class of infinite products, and Euler's totient |
| title_full_unstemmed | A new class of infinite products, and Euler's totient |
| title_short | A new class of infinite products, and Euler's totient |
| title_sort | new class of infinite products and euler s totient |
| topic | combinatorial identities partitions arithmetic functions convergence and divergence of infinite products lattice points in large regions applications of sieve methods combinatorial enumeration problems generating functions. |
| url | http://dx.doi.org/10.1155/S0161171294000591 |
| work_keys_str_mv | AT geoffreybcampbell anewclassofinfiniteproductsandeulerstotient AT geoffreybcampbell newclassofinfiniteproductsandeulerstotient |