One of the formulas is given in [5, Theorem 1.1] and expressed as an infinite sum of which each summand
consists of the modified Bessel functions, the Gegenbauer polynomials and the densities [p.sub.r.sup(0)] (.;x).
where we just take the first summand
in the above infinite series when x is large.
when the elements [t.sub.i,j,l] are considered as elements of the direct summand
A of T or of the direct summand
C of T.
Let us consider the summand
[mathematical expression not reproducible].
leader of the house on Saturday in the Assembly session that already summand
by the Governor Balochistan, Muhammad Kahn Achakzai after receiving the summery sent by the Speaker Raheela Hameed Khan Durrani.Earlier Speaker Balochistan Assembly Wednesday asked Governor Balochistan to call provincial assembly session for electing new Leader of the House.
About Proposition [2.2], we have the following observation: if [mathematical expression not reproducible], when we compute every summand
of [a.sub.i][X.sub.i][b.sub.j][Y.sub.j] we obtain products of the coefficient [a.sub.i] with several evaluations of [b.sub.j] in [sigma]'s and [delta]'s depending of the coordinates of [[alpha].sub.i].
Noting that [xi]1 + [xi]1 [member of] R, we have [1/2]<1 + [[xi].sub.1]+[[xi].sub.1]< [3/2], and therefore the first summand
in (3.2) can be estimated as follows:
Since [mathematical expression not reproducible], the right-hand side of equality (6) and the second summand
of the left-hand side of this equality are divisible by [mathematical expression not reproducible].
For the summand
I1 we integrate by parts via identity
Consequently, the latter is also true for any orthogonal G-representation V without a trivial direct summand
, and it follows that the corresponding vector bundle EG x G V is integrally orientable.
The first summand
describes a comparative efficiency effect.
The last line in the chain of (34) resulted from fixing the random variable [Z.sub.k] in the kth summand
of the second line and then finding statistical expectation with respect to [Z.sub.k] in the kth summand