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Using the above remark, we have also.

Using the above remark we have the following interesting results.

Remark 3.3 Using the above similar ideas, we can obtain similar inequalities of the strictly diagonally M-matrix by column.

Remark 4.2 Using the above methods, one can also derive that for any positive integer n, ∑ i = 0 n ( n i ) ( − q ( x ) ) i k F p, q, ( n − 2 i ) k ( x ) = 0, (4.9).

Remark In fact, using the above formulas, we can also obtain the recurrence formula of the Smarandache-Pascal derived sequence { T n } of { u n }, where { u n } denotes the m th-order linear recursive sequences as follows: u n = a 1 u n − 1 + a 2 u n − 2 + ⋯ + a m − 1 u n − m + 1 + a m u n − m, with initial values u i ∈ N for n > m and 0 ≤ i < m.

Using the above method: 10x/2 = 5x.

The above remark yields the following statement.

From the above remark, we conclude, as.

From the above remark, we have.

Using the remark above and Lemma 4.1, we obtain the following theorem.

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