bibasic sums and series
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21: 33.23 Methods of Computation
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►Cancellation errors increase with increases in and , and may be estimated by comparing the final sum of the series with the largest partial sum.
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22: 36.8 Convergent Series Expansions
§36.8 Convergent Series Expansions
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even,
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odd,
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►For multinomial power series for , see Connor and Curtis (1982).
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23: 7.6 Series Expansions
§7.6 Series Expansions
►§7.6(i) Power Series
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7.6.5
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►The series in this subsection and in §7.6(ii) converge for all finite values of .
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§7.6(ii) Expansions in Series of Spherical Bessel Functions
…24: 3.9 Acceleration of Convergence
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►Similarly for convergent series if we regard the sum as the limit of the sequence of partial sums.
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►If is a convergent series, then
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3.9.2
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►If is the th partial sum of a power series
, then is the Padé approximant (§3.11(iv)).
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25: 31.11 Expansions in Series of Hypergeometric Functions
§31.11 Expansions in Series of Hypergeometric Functions
… ►Such series diverge for Fuchs–Frobenius solutions. …Every Heun function can be represented by a series of Type II. ►§31.11(v) Doubly-Infinite Series
… ►26: 1.15 Summability Methods
27: 27.10 Periodic Number-Theoretic Functions
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►Every function periodic (mod ) can be expressed as a finite Fourier
series of the form
…An example is Ramanujan’s sum:
…It can also be expressed in terms of the Möbius function as a divisor sum:
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►is a periodic function of and has the finite Fourier-series expansion
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is separable for some if
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28: 1.9 Calculus of a Complex Variable
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►Then the series
converges uniformly on .
►A doubly-infinite series
converges (uniformly) on iff each of the series
and converges (uniformly) on .
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►Inside the circle the sum of the series is an analytic function .
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►A double series is the limit of the double sequence
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►If a double series is absolutely convergent, then it is also convergent and its sum is given by either of the repeated sums
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29: 22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial Fractions: Eisenstein Series
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22.12.13