Schr%C3%order%20numbers
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1: 28.12 Definitions and Basic Properties
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►The introduction to the eigenvalues and the functions of general order proceeds as in §§28.2(i), 28.2(ii), and 28.2(iii), except that we now restrict ; equivalently .
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§28.12(ii) Eigenfunctions
… ►For , … ► … ►2: 28.2 Definitions and Basic Properties
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§28.2(vi) Eigenfunctions
…3: 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). …4: 18.42 Software
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►For another listing of Web-accessible software for the functions in this chapter, see GAMS (class C3).
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5: Bibliography S
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A property of the zeros of cross-product Bessel functions of different orders.
Z. Angew. Math. Mech. 56 (2), pp. 120–121.
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Large orders and summability of eigenvalue perturbation theory: A mathematical overview.
Int. J. Quantum Chem. 21, pp. 3–25.
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Asymptotische Entwicklungen der Jacobischen Polynome.
Schr. der König. Gelehr. Gesell. Naturwiss. Kl. 10, pp. 33–112.
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On the derivative of the associated Legendre function of the first kind of integer degree with respect to its order (with applications to the construction of the associated Legendre function of the second kind of integer degree and order).
J. Math. Chem. 46 (1), pp. 231–260.
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On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree (with applications to the construction of the associated Legendre function of the second kind of integer degree and order).
J. Math. Chem. 49 (7), pp. 1436–1477.
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6: 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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►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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…It is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
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§27.2(ii) Tables
…7: 26.14 Permutations: Order Notation
§26.14 Permutations: Order Notation
… ►The set (§26.13) can be viewed as the collection of all ordered lists of elements of : . … ► ►The Eulerian number, denoted , is the number of permutations in with exactly descents. … ►§26.14(iii) Identities
…8: 26.13 Permutations: Cycle Notation
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►In cycle notation, the elements in each cycle are put inside parentheses, ordered so that immediately follows or, if is the last listed element of the cycle, then is the first element of the cycle.
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►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►The derangement number, , is the number of elements of with no fixed points:
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9: 6.16 Mathematical Applications
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§6.16(ii) Number-Theoretic Significance of
►If we assume Riemann’s hypothesis that all nonreal zeros of have real part of (§25.10(i)), then ►
6.16.5
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►where is the number of primes less than or equal to .
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