Download Some multiplication formulas

R ENDICONTI
del
S EMINARIO M ATEMATICO
della
U NIVERSITÀ DI
PADOVA
L EONARD C ARLITZ
Some multiplication formulas
Rendiconti del Seminario Matematico della Università di Padova,
tome 32 (1962), p. 239-242
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SOME MULTIPLICATION FORMULAS
Nota
(*)
di LEONARD CARLITZ
Some years ago Rainville
(a Durham)(**)
[5] proved the formula
where
that
P.(x) is the Legendre polynomial. The writer [11 showed
where
Cf(z) is
For A
the
ultraspherical polynomial defined by
1/2, (2) reduces to (1).
Chatterjea [2] has recently proved
where
=
~"(x)
is defined
the
following
formulas:
by
(*) Pervenuta in redazione il 3 dicembre 1961.
Indirizzo dell’A.: Department of Mathematics, Duke University,
Durham, North Carolina (U.S.A.).
(**) Supported in part by National Science Foundation grant G 16485.
240
and
where L,~(~) is the Laguerre polynomial.
These results and in particular (3) suggest the consideration
of the functional equation
If
we
For
For x
take fl =
=
=
tan
a
it is
easily verified
that
(5) reduces
to
this becomes
1, (6) becomes
where
are arbitrary
Conversely if f n(~1) is defined by (7), where the c,
we have
Indeed
holds.
..constants, it is easily verified that (6)
241
therefore that (6) is equivalent to (7) and that the
of (6) is given by (7) with arbitrary c, . Moreover
solution
general
it follows from (6) with x replaced by pz and A replaced by
that
We
If
see
mow
take A
=
tan a, It
=
tan
~B
we
get (5). Thus (5) is equi-
valent to (7).
We recall that Feldheim [3] has proved that the Laguerre
are the only orthogonal polynomials with
polynomials
a multiplacation theorem of the form
It may be of interest to note that
or, if
with
we
prefer,
(7) implies
242
where the last is symbolic. Rainville [4, p. 244] has noted that
the polynomials f n (~ ) defined by (9) satisfy (6).
It follows from (9) that the polynomials
constitute
an
Appell set, that
is
B~2013//
Also
(6)
becomes
Conversely
polynomials
if the Appell set {g"(x)} is assigned then the
defined by (11) satisfy (5).
REFERENCES
[1] CARLITZ L.: Note
on a formula of Rainville. Bulletin of the Calcutta
Mathematical Society, vol. 51 (1959), pp. 132-133.
[2] CHATTERJEA S. K. : Notes on a formula of Carlitz. Rendiconti del
Seminario Matematica della Università di Padova, vol. 31 (1961),
pp. 243-248.
[3] FELDHEIM E.: Une propriété caracteristique des polynomes de Laguerre. Commentarii Mathematici Helvetici, vol. 13 (1940), pp. 6-10.
[4] RAINVILLE E. D.: Certain generating functions and associated polynomials. American Mathematical Monthly, vol. 52 (1945), pp. 239250.
[5] RAINVILLE E. D.: Notes
American Mathematical
on Legendre polynomials.
Society, vol. 51 (1945), pp.
Bulletin of the
268-271.