SOME FORMULAS OF L . CARLITZ ON HERMITE POLYNOMIALS

We have used the idea of ‘quasi inner product’ introduced by L. R. 
Bragg in 1986 to consider generating series 
∑n=0∞Hn2(x)Hn2(y)tn22n(n!)2 
studied by L. Carlitz in 1963. The pecularity of the series is that there is (n!)2 
in the denominator, which has a striking deviation from the usuaI generating series 
containing n! in the denominator. Our generating function for the said generating 
series is quite different from that of Carlitz, but somewhat analogous to generating 
integrals derived by G. N. Watson (Higher Transcendental function Vol.III, P 271-272 
for the case of Legendre, Gegenbauer and Jacobi polynomials.

In proving the result (1.5) we recall the method of 'quasi inner product', introduced by L.R. Bragg [2], where the name 'quasi inner product' is not appropriate, on account of the fact that inner product has a different meaning, however one may call it simply the product or multiplication like Hadamard's multiplication.
It may be noted that the integrnd is connected with the generating function of the Legendre polynomials.Indeed, using the well-known generating function of the Legendre polynomials and term-by-term integration we obtain f f t2)-1/2 2 I__ Ir(l-2t cos 2{9+ de= -Z tn(f Pn(COS 2e)de).(3.3) The derivation of the particular case (1.10) and (1.11) are similar in nature.
Furthermore, it may be of much interest to compare (1.9) and (1.11) as the left members of two results are apparently different in nature.We now like to examine the left members of (1.9) and (1.11).To do this we first observe that the left member of (1.

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: He has also pointed out that the polynomial n (-1) r H2r(X)H2r(Y) L2 r (2x2+2y2) F (x,y)= r.