By George Gasper

Thorough and updated, this ebook is an authoritative account of uncomplicated hypergeometric sequence and their purposes. The authors offer specific and precise information regarding summation, transformation and growth formulation and contour integrals. They relate the final effects to different vital mathematical features and combinatorial questions.

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**Example text**

10), in analogy with the notation for “f -related’’ bivector ﬁelds on a manifold. Similarly, f : N → M is a backward Dirac map, or simply b-Dirac, if LN = {(Y, df ∗ α) | Y ∈ T N, α ∈ T ∗ M and (df (Y ), α) ∈ LM }. 11) If LM and LN are associated with twisted presymplectic structures ωM and ωN , then a b-Dirac map is just a map satisfying f ∗ ωM = ωN . As before, we will write LN = f ∗ LM to denote that f is a b-Dirac map. Note that f ∗ LM is always a well-deﬁned, though not necessarily smooth, subbundle of T N, in contrast with f∗ LN , which may not be well deﬁned at all.

S. 67). Comparing the two ∗ α replaced by any formulas, we see that the resulting equation makes sense for ρM element in C ∞ (M, g∗ ). On the other hand, since the equation is C ∞ (M)-linear with respect to this element, we may assume that the element is a constant a ∈ g∗ (and the remaining appearances of ρM become ρ). s. 68), becomes −(dµ)(ρ(v), (σ ∨ )∗ a) − Lρ(v) µ, (σ ∨ )∗ a + L(σ ∨ )∗ a µ, ρ(v) = a, [v, σ ∨ µ] , Dirac structures, momentum maps, and quasi-Poisson manifolds or, equivalently, −µ([ρ(v), (σ ∨ )∗ a]) = a, [v, σ ∨ µ] .

65). 65) on an arbitrary 1-form µ ∈ 1 (G). The left-hand side gives J ∗ µ, [ρM (v), π (α)] = d(J ∗ µ)(ρM (v), π (α)) + LρM (v) J ∗ µ, π (α) − Lπ (α) J ∗ µ, ρM (v) ∗ ∗ (α)) − LρM (v) µ, (σ ∨ )∗ ρM (α) = −(dµ)(ρ(v), (σ ∨ )∗ ρM + L(σ ∨ )∗ ρM∗ (α) µ, ρ(v) . 67) Evaluating µ on the right-hand side, we get − LρM (v) (α), ρM σ ∨ µ = −LρM (v) α, ρM σ ∨ µ + α, [ρM (v), ρM σ ∨ µ] . Now, using [ρM (v), ρM (v)] ˜ = ρM ([v, v]) ˜ + ρM LρM (v) (v) ˜ for v˜ = σ ∨ µ ∈ ∞ C (M, g), we get ∗ ∗ ∗ −LρM (v) ρM (α), σ ∨ µ + ρM (α), [v, σ ∨ µ] + ρM (α), LρM (v) (σ ∨ µ) .