SOME FURTHER RESULTS ON LEGENDRE NUMBERS

The Legendre numbers, Pnm, are expressed in terms of those numbers, Pkm−1, in the previous column down to Pnm and in terms of those, Pkm, above but in the same column. Other results are given for numbers close to a given number. The limit of the quotient of two consecutive non-zero numbers in any one column is shown to be −1. Bounds for the Legendre numbers are described by circles centered at the origin. A connection between Legendre numbers and Pascal numbers is exhibited by expressing the Legendre numbers in terms of combinations.


INTRODUCT ION.
The Legendre numbers were introduced in [i] and several elementary properties were given.In [2], some applications of the numbers were presented.Further applications are needed.In this note some relationships between the numbers are shown, bounds are given for the numbers, and the numbers are described in terms of combinations.For reference, we give (from [i]) the definition that we use, a general formula for the numbers, and a partial table of them.Definition i.The Legendre numbers, pm n' are the values of the associated Legendre functions pm(x) for x 0 and m, n non-negative integers.n A general formula for the Legendre numbers is where p(m) (0) is the ruth derivative of the Legendre polynomial, P (x), evaluated n n at x=0.
2. SOME RELATIONSHIPS BETWEEN LEGENDRE NUMBERS.Many relationships between Legendre numbers have been shown in [i], and [2].
Here, each Legendre number is expressed in terms of the non-zero entries in the pre- vious colum (see Table 1) dow to this entry in two ways.Further, each is expressed in terms of the non-zero entries in the same column but above the entry.-9 4 5 4 ----0 10,8_395 0 _135,1352 02,027,025 From the known result, see [3], P'n(X) Z (2n-4k + 3)Pn_2k+l(X (2,1) k=l n n+l where is n is even and ---if n is odd, it follows that by taking m-i derivatives then using (1. Noe that (2.3) and (2.4) give each Legendre number as a product that involves the sum of the absolute values of the entries in the previous column and above the entry of Table I specified.Similarly, (2.5) and (2.6) involve the entries in the same column but above the entry.Equations (2.4) and (2.6) can be obtained from (2.3) and (2.5), repectively, by replacing 2n with 2n + i and 2m with 2m + i.In fact, (2.3) and (2.4) can be comblned as pm (-i) 2(2m-i) Pn-2k+l n k=I while (2.5) and (2.6) can be combined as n-___m for m and n of the same parity.
There are several results concerning entries in Table 1 that are near each other.
These can be easily proved using properties of Legendre numbers or by using (1.1).For example, gives each entry in terms of the entries in the next column and just above and below.
Each entry in terms of the entries in the previous column and just above and below is given by pm (2.10) Considering pm and the nearest entries on slant lines through pm leads to a deter- Pn-iPn+l Pn-iPn+l (n+m-l) (n-m+2) (2.11) Next, if we look at a particular non-zero entry and consider the first four non- zero entries above, below, to the left, and to the right, we have m m pm-2pm+2 nP-2Pn+2n n 0, n _> 4, 2 _< m_< n-2 (2.12) which one can express as above below left right.
In [i], it was shown that the sum of the non-zero entries in any column of Table i converges.The limit of the ratio of consecutive entries is somewhat surprising.
Choose the mth column of Table I.For n + m even and using (i.i), we have pm From (2.13) it is clear that the limit approaches -i from the right for m 0 and from the left for m > i.It is clear that the limit of the absolute value of the ratios is i.

BOUNDS FOR THE LEGENDRE NUMBERS.
From the known bound from [3], In [2], combinations were expressed in terms of the Legendre numbers.Here, we express the Legendre numbers as combinations.The equation C(q,i) i 0 to q (4.1) (q i)!Pq after solving for p + then letting n q + i and m q i.Notice that n-m n+m --i and -q.Since pn n n+m p 2 1-3"5''' (n+m-l) n+m 2 (n + m) n+m for n > 0, m and n of the same parity, and m < n.The remaining values of pm n 0 pm are given in [i] as P0 i and 0 for m and n of different parity.n

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: circumference of a circle of radius n centered at the origin with D 2n the diameter of the circle.In[i], the relationship pm (n+m-l) (n+m-3)-.. (n-m+3) (n-m+l)P n m' m > 1 (3.3) n was given where P is in the first column of Table i.Using (3.2) in (3.3) we have n-m the more general resultIeml < (n+m-l) (n+m-3)-."(n-m+3) (n-re+l)/ n -, m > i, n >m,(3.4)whereC2(n-m) is the circumference of a circle of radius n m centered at the origin with D 2(n-m) the diameter of the circle. 4. LEGENDRE NUMBERS IN TERMS OF COMBINATIONS.