Polynomial of generalized Call

Definition

In Mathematical, a continuation of Polynomial S $\ {p_n (Z) \, \} \,$ have a representation of Appell generalized if generating Fonction of the polynomials takes the form suivante :

K (Z, W) = has (W) \ Psi (zg (W)) = \ sum_ {n=0} ^ \ infty p_n (Z) w^n

where the generating function

K (Z, W) \,

is made up of the series :

$A (W) = \ sum_ {n=0} ^ \ infty a_n w^n \ quad$ with $a_0 \ 0 \,$

$\ Psi (T) = \ sum_ {n=0} ^ \ infty \ Psi_n t^n \ quad$ with all the $\ Psi_n \ 0 \,$

$g (W) = \ sum_ {n=1} ^ \ infty g_n w^n \ quad$ , with $g_1 \ 0.$

Under the conditions above, it is not difficult to show that

p_n (Z) \,

is a polynomial of degree

n \,

Particular cases

the choice of $g (W) =w \,$ gives the class of the polynomials of Brenke .

the choice of $\ Psi (T) =e^t \,$ takes the action pursuant of the polynomials of Sheffer .

the simultaneous choice of $g (W) =w \,$ and of $\ Psi (T) =e^t \,$ in a strict sense takes the action pursuant of the polynomials of Appell .

Explicit representation

The generalized polynomials of Call have the explicit representation

p_n (Z) = \ sum_ {k=0} ^n z^k \ Psi_k h_k.

The coefficient

h_k \,

h_k= \ sum_ {P} a_ {j_0} g_ {j_1} g_ {j_2} \ ldots g_ {j_k}

where the sum extends to all the partitions from N in k+1 left - in the broad sense - i.e. by admitting the part empties for $\ {j_0 \} \,$ ; so that the sum includes/understands all or not the $\ {J \} \,$ , null, such as $j_0+j_1+ \ ldots +j_k = N \,$ . For the polynomials of Call, this becomes the formule :

p_n (Z) = \ sum_ {k=0} ^n \ frac {a_ {n-k} z^k} {K!}

Relations of recurrence

In an equivalent, requirement and sufficient way so that the core

K (Z, W) \,

can be written like

A (W) \ Psi (zg (W))\,

with

g_1=1 \,

is that

\ frac {\ partial K (Z, W)}{\ partial W} =

C (W) K (Z, W) + \ frac {zb (W)}{W} \ frac {\ partial K (Z, W)}{\ partial Z} where

b (W) \,

and

c (W) \,

have a development in series

b (W) = \ frac {W} {G (W)} \ frac {D} {dw} G (W)

1 + \ sum_ {n1} ^ \ infty b_n w^n

and

c (W) = \ frac {1} {has (W)} \ frac {D} {dw} has (W)

\sum_{n0}^\infty c_n w^n.

By making the substitution :

K (Z, W) = \ sum_ {n=0} ^ \ infty p_n (Z) w^n

it comes immediately the Relation from recurrence :

z^ {n+1} \ frac {D} {dz} \ left \ frac {p_n (Z)}{z^n}
\ right] = - \ sum_ {k=0} ^ {n-1} c_ {n-k-1} p_k (Z) - Z \ sum_ {k=1} ^ {n-1} b_ {n-k} \ frac {D} {dz} p_k (Z)

In the particular case of the polynomials of Brenke, there is

g (W) =w \,

and thus all the

b_n=0 \,

, which simplifies the relation of recurrence considerably.

References

Ralph P. Boas, Jr. and R. Creighton Buck, “ Polynomial Expansions off Analytic Functions ” (Corrected 2nd edition), (1964) Academic Close Inc., Publishers, New York, Springer-Verlag, Berlin. Library off Congress Card Number 63-23263.

William C. Brenke, One generating functions off polynomial systems , “ American Mathematical Monthly ”, (1945) 52 pp. 297-301.

W. NR. Huff, standard The off the polynomials generated by F (xt) φ (T) “ Duke Mathematical Journal ”, (1947) 14 p 1091-1104.

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