Euler%20pentagonal%20number%20theorem
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1: 24.1 Special Notation
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Bernoulli Numbers and Polynomials
►The origin of the notation , , is not clear. … ►Euler Numbers and Polynomials
… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations , , as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …2: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►This is the number of positive integers that are relatively prime to ; is Euler’s totient.
►If , then the Euler–Fermat theorem states that
…The
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are relatively prime to and distinct (mod ).
…Note that .
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3: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Application of the combined nonlinear-condensation transformation to problems in statistical analysis and theoretical physics.
Comput. Phys. Comm. 150 (1), pp. 1–20.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Umbral calculus, Bailey chains, and pentagonal number theorems.
J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.
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4: 24.20 Tables
§24.20 Tables
… ►Wagstaff (1978) gives complete prime factorizations of and for and , respectively. …5: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Zhang and Jin (1996, Table 3.8) tabulates for , to 8D or 8S.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
6: 20 Theta Functions
Chapter 20 Theta Functions
…7: 27.14 Unrestricted Partitions
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►Euler introduced the reciprocal of the infinite product
…Euler’s pentagonal number theorem states that
…where the exponents , , , , , , are the pentagonal numbers, defined by
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►Multiplying the power series for with that for and equating coefficients, we obtain the recursion formula
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►This is related to the function in (27.14.2) by
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8: 27.15 Chinese Remainder Theorem
§27.15 Chinese Remainder Theorem
… ►This theorem is employed to increase efficiency in calculating with large numbers by making use of smaller numbers in most of the calculation. …Their product has 20 digits, twice the number of digits in the data. By the Chinese remainder theorem each integer in the data can be uniquely represented by its residues (mod ), (mod ), (mod ), and (mod ), respectively. …These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits. …9: 24.2 Definitions and Generating Functions
§24.2 Definitions and Generating Functions
►§24.2(i) Bernoulli Numbers and Polynomials
… ►§24.2(ii) Euler Numbers and Polynomials
… ►§24.2(iii) Periodic Bernoulli and Euler Functions
… ► …10: 5.22 Tables
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►Abramowitz and Stegun (1964, Chapter 6) tabulates , , , and for to 10D; and for to 10D; , , , , , , , and for to 8–11S; for to 20S.
Zhang and Jin (1996, pp. 67–69 and 72) tabulates , , , , , , , and for to 8D or 8S; for to 51S.
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►Abramov (1960) tabulates for () , () to 6D.
Abramowitz and Stegun (1964, Chapter 6) tabulates for () , () to 12D.
…Zhang and Jin (1996, pp. 70, 71, and 73) tabulates the real and imaginary parts of , , and for , to 8S.