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1: 23.10 Addition Theorems and Other Identities
§23.10(ii) Duplication Formulas
§23.10(iii) n -Tuple Formulas
2: 19.29 Reduction of General Elliptic Integrals
Cubic cases of these formulas are obtained by setting one of the factors in (19.29.3) equal to 1. … (19.29.7) subsumes all 72 formulas in Gradshteyn and Ryzhik (2000, 3.168), and its cubic cases similarly replace the 18 + 36 + 18 = 72 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.142, and 3.141(1-18)). … where e j is an n -tuple with 1 in the j th position and 0’s elsewhere. … The first choice gives a formula that includes the 18+9+18 = 45 formulas in Gradshteyn and Ryzhik (2000, 3.133, 3.156, 3.158), and the second choice includes the 8+8+8+12 = 36 formulas in Gradshteyn and Ryzhik (2000, 3.151, 3.149, 3.137, 3.157) (after setting x 2 = t in some cases). … The first formula replaces (19.14.4)–(19.14.10). …
3: 19.19 Taylor and Related Series
and define the n -tuple 1 2 = ( 1 2 , , 1 2 ) . …
4: 19.18 Derivatives and Differential Equations
Let j = / z j , and e j be an n -tuple with 1 in the j th place and 0’s elsewhere. …
5: 27.20 Methods of Computation: Other Number-Theoretic Functions
The recursion formulas (27.14.6) and (27.14.7) can be used to calculate the partition function p ( n ) for n < N . … A recursion formula obtained by differentiating (27.14.18) can be used to calculate Ramanujan’s function τ ( n ) , and the values can be checked by the congruence (27.14.20). …
6: Howard S. Cohl
Howard is the project leader for the NIST Digital Repository of Mathematical Formulae seeding and development projects. In this regard, he has been exploring mathematical knowledge management and the digital expression of mostly unambiguous context-free full semantic information for mathematical formulae.
7: Preface
Abramowitz and Stegun’s Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables is being completely rewritten with regard to the needs of today. …The authors will review the relevant published literature and produce approximately twice the number of formulas that were contained in the original Handbook. …
8: 5.5 Functional Relations
§5.5(ii) Reflection
5.5.3 Γ ( z ) Γ ( 1 - z ) = π / sin ( π z ) , z 0 , ± 1 , ,
§5.5(iii) Multiplication
Duplication Formula
Gauss’s Multiplication Formula
9: 24.6 Explicit Formulas
§24.6 Explicit Formulas
24.6.6 E 2 n = k = 1 2 n ( - 1 ) k 2 k - 1 ( 2 n + 1 k + 1 ) j = 0 1 2 k - 1 2 ( k j ) ( k - 2 j ) 2 n .
24.6.7 B n ( x ) = k = 0 n 1 k + 1 j = 0 k ( - 1 ) j ( k j ) ( x + j ) n ,
24.6.12 E 2 n = k = 0 2 n 1 2 k j = 0 k ( - 1 ) j ( k j ) ( 1 + 2 j ) 2 n .
10: 27.5 Inversion Formulas
§27.5 Inversion Formulas
which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … Special cases of Möbius inversion pairs are: … Other types of Möbius inversion formulas include: …