From the above, one can also use Theorems 2.2 and 2.3 to establish some similar sum relations to Theorems 2.4 and 2.5.
As further applications, we also give some new sum relations of products of the Frobenius-Euler polynomials by applying some summation transform techniques, which are analogous to some sum relations of products of the classical Euler polynomials and Frobenius-Euler polynomials stated in [9, 25].
In this section, we shall extend the two sum relations due to Lee and Asci and give the close formulae of sums of any positive powers for the ( p, q ) -Fibonacci and Lucas polynomials.
In [11], Lee and Asci applied Property 2.3 to extend a result of Koshy [3] to the ( p, q ) -Fibonacci and Lucas polynomials and give two sum relations for the ( p, q ) -Fibonacci and Lucas polynomials.
This can be expressed as the Arrhenius relationship: S = cK z (Fig. 4 right-hand column) by virtue of the zero-sum relation of N to K (Fig. 4 left-hand column).
Simulations at αij = 0.1 for all i≠j give a zero-sum relation N = 4.026K, which has >4-fold steeper gradient than that for the Lotka-Volterra scenario (Fig. 4) reflecting its >4-fold reduction in α and consistent with its representation of a highly niched scenario.
The company made a net profit of just $274m last year, a minuscule sum in relation to its revenues and its $154-billion value on the stockmarket.
Given there were no clear grounds to sack Barton, it was only the individual's desire to play football elsewhere swiftly that ensured the paying of a paltry sum in relation to the remainder of his two-year contract.
All told shady shelters and good old-fashioned fudging the tax cheating appears to be costing the Treasury well above $200 billion a year, a devastating sum in relation to the $1.1 trillion of income tax collections.
