The special case c− = 0 occurs exactly when ρ 13 = ρ 23 = 0, ρ 12 − = − 1, and (5.3) reduces to the formula C − x, y, z = C 0 x, z + C 0 y, z − 1 2 1 + z, x + y ≥ 0, C − x, y, z = 0, x + y ≤ 0. Inserting the inequalities from Theorem 4.1 (4.2) case by case into the upper (lower) lifting copula (2.14) shows the desired formulas.
Reverting to α≡qρ, we obtain the following desired formula for (5.1) [4].
end{aligned} Combining these two equalities we get the desired formula.
end{aligned} (3.9)The desired formula is obtained by replacing (k) by (-k) on the left-hand side.
end{aligned}Integrating by parts and using (log D zeta )=log a k)+ieta (k)) along with (2.2) yields the desired formula.
end{aligned}The desired formula, (R x)^dagger = R widetilde{psi }(x))), now results from this equation and the identities (T_w^dagger = T_{-w}) and (gamma _x -w) = gamma _x -w).
end{aligned} Using the Wallis formula for π [8], Formula 0.262, we have begin{aligned} sum_{n=1}^{infty} -1)^{n-1} -1biggl(frac{n+1}{n} biggr)&= sum_{n=1}^{infty}ln biggl(frac{2n}{2n-1}cdotfrac {2n}{2n+1} biggr) &=-lnprod_{n=1}^{infty}biggl(1- frac{n+1}{n{2}} biggr)=ln biggl(frac{pi}{2} biggr) end{aligned} and the desired formula follows.
Thus, we have proved the desired formula, D_{r,v}u t,x ={G_{alpha}}(r,v t,x)+ int_{r}^{t}ds int_{mathbb{R}}frac {partial {G_{alpha}}}{partial y}(s,y t,x frac{partial f}{partial z} bigl s,y,u s,y bigr)D_{r,v}u s,y), dy for all (0leq rleq t), (x,vin{mathbb {R}}), and the theorem follows.
We thus obtain u = R f Open image in new window, where f is a holomorphic function in the domain D a, γ, δ Open image in new window, satisfying the following: f ′ ( c a, γ, δ = 1 ζ - 1 1 c a, γ, δ ′ 1 σ S + × P T + ζ - 1 ζ | c a, γ, δ ′ | σ S - 2 U 0, 1 Open image in new window. for all ζ ∈ D Open image in new window. On arguing as in (58), we derive the desired formula for the solution u. □.
The basic idea upon deriving the desired linearization formulas is essentially based upon employing some symbolic computation such as Zeilberger's, Petkovsek's and van Hoeij's algorithms, for the sake of reducing the (_{4}F_{3}(1)), which appears in (9), for certain choices of the involved parameters.
The resulting formula M/(4πr3/3) is then the desired density formula.
