Revista Colombiana de Matemáticas Volumen 32 (1999), páginas 143–172

Sobre el comportamiento asintótico de algunas sucesiones de polinomios

generadas por fórmulas lineales de recurrencia de segundo orden

Yu Takeuchi Universidad Nacional de Colombia, Bogotá

Abstract. The asymptotic behavior of systems of polynomials defined by linear second order recurrence relations of the form

S_{n+1}(z) − (a_{n} · z + b_{n}) · S_{n}(z) + c_{n} · S_{n−1}(z) = 0,

where a_{n} → 2, b_{n} → 0, c_{n} → 1, is studied. As an application of the results, the domains of convergence of polynomial series of the form

∑_{n=0}^{∞}p_{n}S_{n} (z) , p_{n} ∈ ℂ,

with lim sup \sqrt{|p_{n}|} = \frac{1}{ρ} < 1 are determined, and the well known Abel’s lemma for power series is extended to such series under appropriate assumptions. The latter problem had been previously addressed by professor Miguel Dumett in [3] from a different point of view. Key words and phrases. Linear second order recurrence relations, independent solutions, asymptotic behavior of solutions, bounded and uniformly bounded variation, Chebyshev, Legendre, Gegenbauer and Jacobi polynomials, Abel-Picard convergence.

1991 Mathematics Subject Classification. Primary 42C05. Secondary 42C10,

42C15. Resumen. En el presente trabajo se estudia el comportamiento asintótico de sucesiones de polinómios generadas por fórmulas lineales de recurrencia de segundo orden de la forma

S_{n+1}(z) − (a_{n} · z + b_{n}) · S_{n}(z) + c_{n} · S_{n−1}(z) = 0,

donde a_{n} → 2, b_{n} → 0, c_{n} → 1. Como aplicación de los resultados se investiga el dominio de convergencia de las series de polinomios

∑_{n=0}^{∞}p_{n}S_{n} (z) , p_{n} ∈ ℂ,

cuando lim sup \sqrt[n]{|p_{n}|} = \frac{1}{ρ} < 1, y se discute la validez para estas series de un análogo del teorema del límite de Abel-Picard, válido para las series de potencias. Este último problema fue considerado previamente por el profesor Miguel Dumett en [3] desde un punto de vista diferente.

Introducción

Así como muchas funciones clásicas de la matemática, las hipergeométricas entre ellas, están relacionadas con las soluciones de ecuaciones diferenciales lineales de segundo orden, especialmente con aquellas que definen relaciones de ortogonalidad (la de Bessel, por ejemplo), muchas otras funciones igualmente importantes, entre otras los sistemas clásicos de polinomios ortogonales (Chebyshev, Legendre, Laguerre, Hermite, etc.), están relacionados con fórmulas lineales de recurrencia de segundo orden. Y así como los desarrollos en funciones ortogonales (Bessel, por ejemplo) juegan papeles fundamentales en la teoría general de las ecuaciones diferenciales y en sus aplicaciones, lo mismo es cierto de estas últimas funciones, especialmente en áreas como el análisis numérico y la teoría de la aproximación.

Aunque la determinación explícita de las soluciones de una relación de recurrencia no es en general una tarea fácil, investigaciones relativamente recientes han mostrado que muchas propiedades de éstas pueden obtenerse directamente a partir del exámen de los coeficientes de dicha relación. Esto es particularmente cierto del comportamiento asintótico de las soluciones. Conocer el comportamiento asintótico de eśtas es especialmente útil para establecer las propiedades de convergencia de los desarrollos en series de dichas soluciones. Curiosamente, aun si las soluciones explícitas son conocidas, el comportamiento asintótico de las mismas puede todavía suministrar valiosa información. El objeto de este trabajo es mostrar cómo este análisis puede realizarse en algunos casos típicos, que incluyen los polinomios de Chebyshev y Legendre, y mostrar cómo tal información puede usarse para examinar la convergencia de los desarrollos propios correspondientes, especialmente en lo relacionado con las propiedades de tales desarrollos en puntos de la frontera de sus dominios de convergencia.

1. Fórmulas lineales de recurrencia de segundo orden

1.1. Transformación de los coeficientes.

La fórmula lineal de recurrencia de segundo orden tiene la forma general

X̂_{n+1} − â_{n} · X̂_{n} + ĉ_{n} · X̂_{n−1} = 0, n = 1, 2, 3, …  (1.0)

Nosotros examinaremos (1.0) bajo hipótesis que cobijan muchos de los casos más significativos:

â_{n} → a ≠ 0, ĉ_{n} → c.  (1.1)

Mediante el cambio de variables

X̂_{n} = ∏^{n−1} ; k=1 \frac{â_{k}}{a} · X_{n}, n ⩾ 2, X̂_{1} = X_{1}, X̂_{0} = X_{0},

(1.0) se transforma en

X_{n+1} − a · X_{n} + c_{n−1} · X_{n−1} = 0, n = 1, 2, 3, …  (1.2)

donde

c_{1} = \frac{a}{â_{1}} · ĉ_{1}, c_{n} = \frac{a^{2}}{â_{n−1}â_{n}} · ĉ_{n} → c.  (1.3)

En esta sección estudiaremos (1.0) en la presentación (1.2).

1.2. Comportamiento asintótico de las soluciones.

Dividiendo (1.2) por X_{n} y haciendo T_{n} = X_{n}/X_{n−1}, n = 1, 2, 3, …, se obtiene la siguiente fórmula fraccionaria de recurrencia de primer orden para (T_{n}) :

T_{n+1} = a − \frac{c_{n}}{T_{n}} , c_{n} → c, n = 1, 2, 3, …  (1.4)

Supongamos que a^{2} − 4c > 0. Entonces, la ecuación cuadrática t^{2} − at + c = 0, llamada la ecuación característica de la fórmula de recurrencia (1.2), posee dos raíces diferentes λ, μ (|λ| < |μ|). Sabemos de [10], [12] que existe p tal que

T_{n} = \frac{X_{n}}{X_{n−1}} → μ si y sólo si T_{1} = \frac{X_{1}}{X_{0}} ≠ p,
T_{n} = \frac{X_{n}}{X_{n−1}} → λ si y sólo si T_{1} = \frac{X_{1}}{X_{0}} = p

(1.5)

Del segundo teorema de Cauchy (veánse [2], [8]) se tiene que

lim_{n→∞} \sqrt[n]{|X_{n}|} = |μ| si y sólo si \frac{X_{1}}{X_{0}} ≠ p,
lim_{n→∞} \sqrt[n]{|X_{n}|} = |λ| si y sólo si \frac{X_{1}}{X_{0}} = p.

(1.6)

Esto conduce a dos tipos diferentes de soluciones de la fórmula de recurrencia (1.2). En el primer caso diremos que (X_{n}) es la solución del tipo (μ^{n}); en el segundo, que (X_{n}) es del tipo (λ^{n}). Denotaremos esta última solución con (Y_{n}). Estas soluciones tienen respectivamente las siguientes expresiones:

X_{n} = α_{n} · μ^{n}, con lim_{n→∞} \sqrt[n]{|α_{n}|} = 1,
Y_{n} = β_{n} · λ^{n}, con lim_{n→∞} \sqrt[n]{|β_{n}|} = 1

(1.7)

Usando la notación “ ∼ ” para denotar la igualdad asintótica, podemos escribir

X_{n} ∼ α_{n} · μ^{n}, Y_{n} ∼ β_{n} · λ^{n}, con \sqrt[n]{|α_{n}|} ∼ 1, \sqrt{|β_{n}|} ∼ 1.  (1.8)

Se ve inmediatamente que lim_{n→∞} Y_{n}/X_{n} = 0. Si (Y_{n}), (Ỹ_{n}) son ambas soluciones del tipo (λ^{n}), entonces Y_{1}/Y_{0} = Ỹ_{1}/Ỹ_{0} = p. Por lo tanto, las soluciones del tipo (λ^{n}) son linealmente dependientes y forman un espacio lineal de dimensión 1 (véase [6]). Por otra parte, si (X_{n}) y (X̃_{n}) son soluciones de tipo (μ^{n}), existe una constante A ≠ 0 tal que X̃_{n} ∼ A · X_{n}. En efecto, (X̃_{n}) es una combinación lineal de (X_{n}) y (Y_{n}) donde (Y_{n}) es solución de (1.3) del tipo (λ^{n}), y existen entonces constantes A y B, A ≠ 0, tales que

(X̃_{n}) = A · (X_{n}) + B · (Y_{n}).

Entonces

\frac{X̃_{n}}{X_{n}} = A + B · \frac{Y_{n}}{X_{n}} ⟶ A, n → ∞.

Teorema 1.1 (Criterio de comparación). Considérense las fórmulas lineales de recurrencia

\begin{matrix}X_{n+1} − a · X_{n} + c_{n} · X_{n−1} = 0, \\ X̃_{n+1} − a · X̃_{n} + c̃_{n} · X̃_{n−1} = 0, \end{matrix}  (1.9)

donde

c_{n} → c, c̃_{n} → c.  (1.10)

Supóngase además que

∞ ; ∑_{n=1} |c_{n} − c̃_{n}| < +∞.  (1.11)

Entonces (i) Si (X_{n}), (X̃_{n}) son soluciones respectivas de las dos fórmulas (1.9), y son

ambas del mismo tipo (μ^{n}), existe una constante C ≠ 0 tal que X̃_{n} ∼ C · X_{n}; esto es,

lim_{n→∞} \frac{X̃_{n}}{X_{n}} = C.  (1.12)

(ii) Si (Y_{n}), (Ỹ_{n})son soluciones respectivas de las dos fórmulas (1.9), y son ambas del mismo tipo (λ^{n}), existe una constante D ≠ 0 tal que Ỹ_{n} ∼ D · Y_{n}; esto es,

lim_{n→∞} \frac{Ỹ_{n}}{Y_{n}} = D.  (1.13)

Demostración. (i) Sean

μ_{n} = \frac{X_{n}}{X_{n−1}} , μ̃_{n} = \frac{X̃_{n}}{X̃_{n−1}}

Entonces (μ_{n}) , \binom{}{μ_{n}} satisfacen

μ_{n+1} = a − \frac{c_{n}}{μ_{n}} , μ̃_{n+1} = a − \frac{c̃_{n}}{μ̃_{n}} , μ_{n} → μ, μ̃_{n} → μ,  (1.14)

y de (1.14) se obtiene inmediatamente que

(μ̃_{n+1} − μ_{n+1}) = \frac{c̃_{n}}{μ_{n}μ̃_{n}} · (μ̃_{n} − μ_{n}) + \frac{c_{n} − c̃_{n}}{μ_{n}} , n = 1, 2, 3, …,

la cual es una fórmula lineal de recurrencia del primer orden para la sucesión (μ̃_{n} − μ_{n}) . Por la hipótesis (1.11) se tiene que

∑_{n=1}^{∞}|\frac{c_{n−}c̃_{n}}{μ_{n}} | < +∞.

Además,

\frac{c̃_{n}}{μ_{n}μ̃_{n}} ⟶ \frac{c}{μ^{2}} = \frac{λ · μ}{μ^{2}} = \frac{λ}{μ}, |\frac{λ}{μ}| < 1.

Por lo tanto (véase el Apendice A),

∑_{n−1}^{∞} |μ̃_{n} − μ_{n}| < +∞,  (1.15)

y como

X_{n} = X_{0} · ∏_{k=1}^{n} \frac{X_{k}}{X_{k−1}} = X_{0} · ∏_{k=1}^{n} μ_{k},  X̃_{n} = X̃_{0} · ∏_{k=1}^{n} μ̃_{k},

entonces

\frac{X̃_{n}}{X_{n}} = \frac{X̃_{0}}{X_{0}} · ∏_{k=1}^{n} \frac{μ̃_{k}}{μ_{k}} = \frac{X̃_{0}}{X_{0}} · ∏_{k=1}^{n} [1 + \frac{μ̃_{k} − μ_{k}}{μ_{k}} ] ,  (1.16)

siendo el producto convergente en virtud de (1.15). Se deduce que

lim_{n→∞} \frac{X̃_{n}}{X_{n}}

existe y es no nulo.

(ii) La demostración de (ii) es similar a la de (i). □ ✓

Comparando la fórmula (1.2) con la fórmula lineal de recurrencia

X_{n+1} − a · X_{n} + c · X_{n−1} = 0, n = 1, 2, 3 …,  (1.17)

la cual tiene coeficientes constantes, y teniendo en cuenta que las progresiones geométricas (μ^{n}; n = 0, 1, 2, …) y (λ^{n}; n = 0, 1, 2, …) son ambas soluciones de (1.17), se obtiene el siguiente colorario.

Corolario 1.1. Si en la fórmula lineal de recurrencia (1.2) suponemos que

∞ ; ∑_{n=1} |c_{n} − c| < +∞,  (1.18)

y si (X_{n}), (Y_{n}) son soluciones de los tipos (μ^{n}) y (λ^{n}), respectivamente, entonces existen constantes α ≠ 0 y β ≠ 0 tales que X_{n} ∼ α · μ^{n}, Y_{n} ∼ β · λ^{n}.

Como cualquier solución de la fórmula lineal de recurrencia (1.2) es una combinación lineal de las dos soluciones (X_{n}) y (Y_{n}) mencionadas en el corolario anterior, se obtiene también el siguiente corolario.

Corolario 1.2. Si (X_{n}) es una solución no trivial de la fórmula de recurrencia (1.2), entonces lim_{n→∞} X_{n}/μ^{n} existe. Además, lim_{n→∞} X_{n}/μ^{n} = 0 si y sólo si lim_{n→∞} X_{n}/λ^{n} existe y es no nulo.

Ejemplo 1.1. Consideremos la fórmula lineal de recurrencia

X_{n+1} − 3 · X_{n} + \frac{2n(n + 3)}{(n + 1)(n + 2)} · X_{n−1} = 0, n = 1, 2, 3, …,

así que

a = 3, c_{n} = \frac{2n(n + 3)}{(n + 1)(n + 2)} ⟶ c = 2, λ = 1, μ = 2.

Además,

∑_{n=1}^{∞} |c_{n} − c| = ∑_{n=1}^{∞} \frac{4}{(n + 1) (n + 2)} = 2.

Según el Corolario 1.1, existen entonces soluciones (X_{n}) , (Y_{n}) de tal fórmula de recurrencia tales que X_{n} ∼ 2^{n}, Y_{n} ∼ 1. En efecto, un cálculo directo muestra que la sucesión (X_{n}) = 2^{n} · (n − 1)/(n + 1) es una solución. Una segunda solución (Y_{n}) está dada por

Y_{n} = X_{n} · ∑_{k=n+1}^{∞} \frac{c_{1}c_{2···}c_{k−1}}{X_{k−1}X_{k}} = \frac{n − 1}{n + 1} · \frac{2^{n}}{3} · ∑_{k=n+1}^{∞} \frac{(k + 1) (k + 2)}{(k − 2)(k − 1)2^{k}} ∼ \frac{1}{3},

como resulta de la Regla de L’Hóspital en [8].

Si la hipótesis (1.18) no se satisface, el siguiente teorema da aún el comportamiento asintótico de las soluciones de (1.2). Usaremos las siguientes notaciones:

σ = 1 ó σ = −1; ε_{n} = c_{n} − c, n = 1, 2, 3, …

Teorema 1.2. En la fórmula de recurrencia (1.2), supóngase que

1. La sucesión (σ^{n} · ε_{n}; n = 1, 2, 3, …) es de variación acotada; esto es, supóngase que

∑_{n=1}^{∞} |ε_{n+1} − σ · ε_{n}| < +∞.

2. También,

∑_{n=1}^{∞} |ε_{n}|^{2} = ∑_{n=1}^{∞} |c_{n} − c|^{2} < +∞.

Entonces

(i) Si (X_{n}) es una solución de (1.2) del tipo (μ^{n}), existe una constante C ≠ 0 tal que

X_{n} ∼ C · μ^{n} · ∏_{k=1}^{n} (1 + \frac{1}{μ(λ − σμ)} · ε_{k}) .

(ii) Si (Y_{n}) es una solución de (1.2) del tipo (λ^{n}), existe una constante D ≠ 0 tal que

Y_{n} ∼ D · λ^{n} ∏_{k=1}^{n} (1 + \frac{1}{λ (μ − σλ)} · ε_{k}) .

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Demostración. (i) Sea (X̃_{n}) dada por

X̃_{n} = μ^{n} · ∏_{k=1}^{n} (1 + t · ε_{k}) , t = \frac{1}{μ (λ − σμ)}.

Un cálculo directo muestra que (X̃_{n}) satisface la fórmula lineal de recurrencia

X̃_{n+1} − a · X̃_{n} + c̃_{n} · X̃_{n−1} = 0, n = 1, 2, 3, …,

donde

c̃_{n} = c + λμ · t · ε_{n} − μ^{2}t · (1 + t · ε_{n}) · ε_{n+1}
= c + t · μ(λ − σμ) · ε_{n} + O(ε_{n}^{2}) + O(Δ_{n})
= c + ε_{n} + O(ε_{n}^{2}) + O(Δ_{n}).

y Δ_{n} = ε_{n+1} − σ · ε_{n}.

De las hipótesis 1 y 2 se obtiene que ∑_{n=1}^{∞} |c̃_{n} − c_{n}| < +∞, y la igualdad asintótica para la solución (X_{n}) resulta entonces del Teorema 1.1. (ii) La demostración de (ii) es similar a la de (i). □ ✓ Ejemplo 1.2. Considérese la fórmula lineal de recurrencia

X_{n+1} − 3 · X_{n} + \frac{2n}{n + 1} · X_{n−1} = 0, n = 1, 2, 3, …

así que

a = 3, c_{n} = \frac{2n}{n + 1} = 2 − \frac{2}{n + 1}, ε_{n} = −\frac{2}{n + 1}, λ = 1, μ = 2.

Claramente

∑_{n=1}^{∞} ε^{2} < +∞, ∑_{n=1}^{∞} |ε_{n+1} − ε_{n}| = ∑_{n=1}^{∞} \frac{2}{(n + 1) (n + 2)} < +∞ (σ = 1) .

En virtud del Teorema 1.2, existen entonces soluciones (X_{n}) , (Y_{n}) tales que

X_{n} ∼ 2^{n} ∏_{k=1}^{n} (1 + \frac{1}{k + 1}) = \frac{n + 2}{2} · 2^{n}, Y_{n} ∼ ∏_{k=2}^{n} (1 − \frac{2}{k + 1}) = \frac{2}{n (n + 1)}.

Ejemplo 1.3. Considérese la fórmula lineal de recurrencia

X_{n+1} − 3 · X_{n} + (2 + \frac{(−1)^{n}}{n} ) · X_{n−1} = 0, n = 1, 2, 3, …,

en la cual

a = 3, c_{n} = 2 + \frac{(−1)^{n}}{n} , ε_{n} \frac{(−1)^{n}}{n} , λ = 1, μ = 2.

Se observa que

∑_{n=1}^{∞}ε_{n}^{2} = ∑_{n=1}^{∞} \frac{1}{n^{2}} < +∞, ∑_{n=1}^{∞} |ε_{n} + ε_{n+1}| = ∑_{n=1}^{∞} \frac{1}{n(n + 1)} < +∞.

(En este caso σ = −1). El Teorema 1.2 asegura entonces la existencia de soluciones (X_{n}) , (Y_{n}) tales que

X_{n} ∼ 2^{n} · ∏_{k=1}^{n} (1 + \frac{(−1)^{k}}{6k} ) , Y_{n} ∼ ∏_{k=1}^{n} (1 + \frac{(−1)^{k}}{3k} ) .

Como los productos infinitos convergen (véase [11]), se concluye que X_{n} ∼ C · 2^{n}, Y_{n} ∼ D , donde C y D son constantes no nulas.

1.3. Una aplicación del método de variación de parámetros.

Si λ, μ (|λ| ⩽ |μ|) son las raíces de la ecuación característica de la fórmula de recurrencia (1.2), tenemos el siguiente teorema (la restricción a^{2} − 4c > 0 impuesta en el parágrafo anterior no es necesaria). Teorema 1.3 Si (X_{n}) es cualquier solución de la fórmula (1.2), entonces

lim sup \sqrt[n]{|X_{n}|} ⩽ |μ| .  (1.18)

Demostración. Escribiendo (1.2) en la forma

X_{n+1} − a · X_{n} + c · X_{n−1} = ε_{n} · X_{n−1},  (1.19)

donde ε_{n} = c − c_{n}, y aplicando el “método de variación de parámetros” (Apéndice B) a la fórmula (1.19), se obtiene que:

(i) Cuando λ ≠ μ (esto es, cuando a^{2} − 4c ≠ 0),

X_{n} = A · μ^{n} + B · λ^{n} + \frac{1}{μ − λ} · ∑_{k=1}^{n} (μ^{n−k} − λ^{n−k}) · ε_{k} · X_{k−1}, n = 1, 2, 3, …, (1.20)

donde

A = \frac{X_{1} − λX_{0}}{μ − λ} , B = \frac{−X_{1} + μX_{0}}{μ − λ} .

Dividiendo (1.20) por μ^{n}, teniendo en cuenta que |μ^{n−k} − λ^{n−k}| ⩽ 2 · |μ^{n−k}| ,

\begin{aligned}|\frac{X_{n}}{μ^{n}} | & ≤ |A| + |B| + \frac{2}{|μ (μ − λ)|} ∑_{k−1}^{n} |ε_{k}| |\frac{X_{k−1}}{μ^{k−1}} | \\ & < M + ∑_{k=1}^{n} d_{k} · |\frac{X_{k−1}}{μ^{k−1}} | , \end{aligned}  (1.21)

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donde

M = |A| + |B| ⩾ |A + B| = |X_{0}| , d_{k} = \frac{2}{|μ (μ − λ)|} · |ε_{k}| ,

y de (1.21) (Apéndice C),

|\frac{X_{n}}{μ^{n}} | ⩽ M · ∏_{k=1}^{n} (1 + d_{k}) .  (1.22)

Como d_{k} → 0 cuando k → ∞, (1.22) implica (1.18) (ii) Cuando λ = μ (esto, es cuando a^{2} − 4c = 0),

X_{n} = μ^{n−1} (nX_{1} − (n − 1) μX_{0}) + ∑_{k=1}^{n−1} (n − k) · μ^{n−k−1} · ε_{k}X_{k−1}, (1.23)

y si r > |μ| es arbitrario , dividiendo (1.23) por r^{n} se concluye que

|\frac{X_{n}}{r^{n}} | ⩽ (\frac{|μ|}{r})^{n} |\frac{nX_{1} − (n − 1) μX_{0}}{μ} | + ∑_{k=1}^{n} (\frac{|μ|}{r})^{n−k} · \frac{n − k}{|μ| · r} · |ε_{k}| |\frac{X_{k−1}}{r^{k−1}} | .

Como |μ|/r < 1, existirá entonces M_{0} > 0 tal que

(\frac{|μ|}{r})^{n} · |\frac{nX_{1} − (n − 1) μX_{0}}{μ} | ⩽ M_{0}, (\frac{|μ|}{r})^{n} · \frac{n}{|μ| · r} ⩽ M_{0}

para todo n. Razonando en forma análoga al caso (i) se obtiene entonces, en lugar de (1.22), que

|\frac{X_{n}}{r^{n}} | ⩽ M_{0} · ∏_{k=1}^{n} (1 + M_{0} |ε_{k}|) .

Por lo tanto, lim sup \sqrt[n]{|X_{n}|} ⩽ r. Como r > |μ| es arbitratio, la relación anterior implica (1.18) □ ✓ Nota 1.1. Si en el caso a^{2} −4c ≠ 0, ∑_{n=1}^{∞} |ε_{n}| < +∞, entonces ∏_{k=1}^{∞} (1 + d_{k}) converge absolutamente. En consecuencia, M · ∏_{k=1}^{∞} (1 + d_{k}) es una cota superior de la sucesión (|X_{n}/μ^{n}|; n = 0, 1, 2, …) .

2. Sucesiones de polinomios

2.1. Los polinomios de Chebyshev T_{n} (z) , n = 0, 1, 2, … La sucesión (T_{n} (z) ; n = 0, 1, 2, …) es la solución de la fórmula lineal de recurrencia de segundo orden

T_{n+1} (z) − 2z · T_{n} (z) + T_{n−1} (z) = 0, n = 1, 2, 3, …  (2.0)

que satisface las condiciones iniciales

T_{0} (z) = 1, T_{1} (z) = z.  (2.1)

Evidentemente T_{n}(z) es un polinomio de grado n en la variable z, que suponemos que toma valores complejos.

Como la fórmula de recurencia (2.0) es lineal de segundo orden, con coeficientes constantes, su solución general se puede encontrar inmediatamente. Sean λ (z) , μ (z) las raíces de su ecuación característica t^{2} − 2z · t + 1 = 0. Entonces

λ (z) , μ (z) = z ± \sqrt{z^{2} − 1}.  (2.2)

Teniendo ahora en cuenta que las progresiones geométricas (λ(z)^{n}; n = 0, 1, 2 …), (μ(z)^{n}; n = 0, 1, 2 …) son soluciones linealmente independientes de (2.0) , consideraremos separadamente tres casos:

(i) z = x (real), −1 < x < 1. Entonces

λ (x) , μ (x) = x ± i · \sqrt{1 − x^{2}} = e^{±iθ}, x = cos θ, 0 < θ < π, (2.3)

así que |λ (z)| = |μ (z)| = 1 y T_{n}(x) = cos nθ. Evidentemente la sucesión T_{n} (x) /T_{n−1}(x) = cos nθ/ cos(n − 1)θ diverge cuando n → ∞. Sin embargo, se tiene (Teorema 1.3) que

lim sup \sqrt[n]{|T_{n} (x)|} = lim sup \sqrt[n]{|cos nθ|} ⩽ 1, −1 < x < 1.

Nótese que lim_{n→∞} \sqrt[n]{|T_{n}(x)|} no siempre existe y que el conjunto

\{x ∈ (−1, 1) : lim inf \sqrt[n]{|T_{n}(x)|} = 0\}

es no vacío y de medida de Lebesgue nula (véase [7]).

(ii) x = ±1. En este caso λ = μ = x = ±1, |λ (±1)| = |μ (±1)| = 1. Tenemos además que T_{n} (x) = x^{n} y que

lim_{n→∞} \frac{T_{n} (x)}{T_{n−1} (x)} = x = ±1.

En consecuencia, lim_{n→∞} \sqrt[n]{|T_{n} (x)|} = 1.

(iii) z ∈ Ω = ℂ ∖ [−1, 1]. Como λ (z) , μ (z) = z ± \sqrt{z^{2} − 1,} (si + corresponde a λ (z), lo hará a μ (z), y recíprocamente), un cálculo elemental muestra que |λ (z)| ≠ |μ (z)| para todo z en la región Ω. Para z ∈ Ω se escogen ahora λ (z) , μ (z) de tal manera que

|λ (z)| < |μ (z)| .  (2.4)

Nótese que |λ (z)| < 1 < |μ (z)| (puesto que |λ (z)| · |μ (z)| = 1).

Cuando z = x real, |x| > 1, teniendo en cuenta que √ representa el valor positivo de la raíz, la escogencia anterior implica que

μ (x) = x + \sqrt{x^{2} − 1}, λ (x) = x − \sqrt{x^{2} − 1}, para x > 1,
μ (x) = x − \sqrt{x^{2} − 1}, λ (x) = x + \sqrt{x^{2} − 1}, para x < −1.

Es bien conocido que λ (z) , μ (z) , definidas por (2.2) y (2.4), son funciones analíticas en Ω con λ′ (z) ≠ 0, μ′ (z) ≠ 0, z ∈ Ω.

Ahora, T_{n} (z) es una combinación lineal de λ (z)^{n} , μ (z)^{n}; esto es,

T_{n} (z) = A · λ (z)^{n} + B · μ (z)^{n} ,

donde A y B son constantes. Como además T_{0} (z) = 1 y T_{1} (z) = z, entonces A = B = \frac{1}{2}, y por lo tanto

\begin{aligned}T_{n} (z) & = \frac{1}{2} · (λ (z)^{n} + μ (z)^{n}) \\ & = \frac{1}{2} · ((z + \sqrt{z^{2} − 1})^{n} + (z − \sqrt{z^{2} − 1})^{n}) . \end{aligned}  (2.5)

De (2.5) se obtiene entonces, para z ∈ Ω, la expresión asintótica

T_{n} (z) = \frac{μ (z)^{n}}{2} · (1 + (\frac{λ (z)}{μ (z)})^{n}) ∼ \frac{1}{2} · μ (z)^{n} , cuando n → ∞,

Por lo tanto, lim_{n→∞} T_{n} (z) /T_{n−1} (z) = μ (z) y, en consecuencia,

lim_{n→∞} \sqrt[n]{|T_{n} (z)|} = |μ (z)| , z ∈ Ω.  (2.6)

2.2. Series de polinomios de Chebyshev.

Como es claro, la serie

∑_{n=0}^{∞}p_{n} · T_{n} (z) , p_{n} ∈ ℂ,  (2.7)

converge si lim sup \sqrt[n]{|p_{n} · T_{n} (z)|} < 1 y diverge si lim sup \sqrt[n]{|p_{n} · T_{n} (z)|} > 1. Para z ∈ Ω, (2.6) implica que

lim sup \sqrt[n]{|p_{n} · T_{n} (z)|} = lim sup \sqrt[n]{|p_{n}|} · lim \sqrt[n]{|T_{n} (z)|}
= |μ (z)| · lim sup \sqrt[n]{p_{n}}.

Por lo tanto, puesto que |μ(z)| > 1, si lim sup \sqrt[n]{|p_{n}|} ⩾ 1 entonces lim sup \sqrt[n]{|p_{n} · T_{n} (z)|} > 1 para todo z ∈ Ω. Por esta razón supondremos, de ahora en adelante, que

\frac{1}{ρ} = lim sup \sqrt[n]{|p_{n}|} < 1.  (2.8)

En tal caso, la serie (2.7) converge absolutamente cuando |μ (z)| < ρ y diverge cuando |μ (z)| > ρ. Nótese que (2.7) converge para z ∈ [−1, 1] .

Con el fin de investigar para qué valores de z ∈ ℂ la serie (2.7) converge es conveniente hacer uso de coordenadas elípticas en el plano. Denotándolas con (s, t), éstas están relacionadas con las coordenadas cartesianas (x, y) por las ecuaciones

x = cos s · cosh t, y = sens · senh t; s ∈ [0, 2π] , t ∈ [0, ∞] .  (2.9)

Se observa que para t fijo, el punto z = x + iy describe la elipse

\frac{x^{2}}{cosh^{2} t} + \frac{y^{2}}{senh^{2}t} = 1.  (2.10)

A su vez, para s fijo, z = x + iy describe la hipérbola

\frac{x^{2}}{cos^{2} s} − \frac{y^{2}}{sen^{2}s} = 1.  (2.11)

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 x y  s fijo  t fijo ↖  ↖ • •_{−1}  _{1}^{0}

Figura 1

Estas curvas se intersectan ortogonalmente y los focos de ambas están localizados en los puntos −1, 1 (Figura 1).

De (2.9) obtenemos también que

z = x + iy = cos s · cosh t + i · sen s · senht = cos (s − it) ,

así que

\sqrt{z^{2} − 1} = i · \sqrt{1 − z^{2}} = i · \sqrt{1 − cos^{2} (s − it)} = ± i sen (s − it) .

Por lo tanto,

λ (z) , μ (z) = z ± \sqrt{z^{2} − 1} = cos (s − it) ± i sen (s − it)
= e^{t} · e^{is}, ó e^{−t} · e^{−is}

Teniendo ahora en cuenta que |λ (z)| < 1 < |μ (z)|, y t ≥ 0, se concluye que

λ (z) = e^{−t} · e^{−is}, μ (z) = e^{t} · e^{is}, z ∈ Ω.  (2.12)

Si z satisface a la ecuación |μ (z)| = ρ entonces t = log ρ. En este caso el punto z = x + iy describe la elipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,  (2.13)

donde

a = cosh (log ρ) = \frac{1}{2} · (ρ + \frac{1}{ρ}) , b = sinh (log ρ) = \frac{1}{2} · (ρ − \frac{1}{ρ}) .

Evidentemente |μ (z)| < ρ si y sólo si z está en el interior de la elipse (2.13), y |μ (z)| > ρ si y sólo si z está en el exterior de la misma. Por lo tanto, si z está en el interior de la elipse (2.13), la serie (2.7) converge absolutamente, mientras que si z está en el exterior de dicha elipse, la serie diverge.

En otras palabras, la elipse (2.13) es “la elipse de convergencia” de la serie (2.7) de polinomios de Chebyshev.

Teorema 2.1. (Teorema del límite de Abel para las series de polinomios de Chebyshev). Sea z_{0} un punto sobre la elipse de convergencia (2.13), esto es, |μ (z_{0})| = ρ. Si la serie (2.7) converge en z_{0}, entonces

∑_{k=0}^{∞}p_{k} · T_{k}(z) ⟶ ∑_{k=0}^{∞} p_{k} · T_{k} (z_{0})  (2.14)

cuando z, con |μ (z)| < ρ, tiende a z_{0} sobre cualquier curva regular que corte a la elipse (2.13) en z_{0} y forme con ésta un ángulo de corte φ no nulo.

Demostración. Considérese la representación conforme w = μ (z), con μ(z) dada por (2.3) y (2.4) (Figura 2).

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.......................................................................................................................................................  x y  • •_{−1}  _{1}_{_{[}[_{[}}  _{_{]}]_{]}}^{^{(}^{(}^{(}}  a b φ K z  z_{o} plano z  elipse |μ(z)| = ρ_{.}_{.}^{.}_{.}_{.}^{.}^{.}_{.}_{.}_{.}^{.}^{.} _{.}_{.}_{.}_{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.}^{.}_{.}_{.}_{.}_{.}^{.}^{.}^{.} _{.}_{.}_{.}^{.}^{.}_{.}_{.}^{.}^{.}_{.}_{.}^{.} ................................................................................................................................................................ ^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} w = μ(z)^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}  ^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}  _{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.}_{.} 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..............................................................................................................................................................  x y  • • plano w μ(K) círculo |w| = ρ φ w  ρ_{1} 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Figura 2

Mediante esta representación, la región {z : |μ (z)| < ρ, z ∉ [−1, 1]} del plano z se transforma en el ánulo {w : 1 < |w| < ρ} del plano w. Si K es una curva regular que corta a la elipse en z_{0} con un ángulo α ≠ 0, μ(K) es también una curva regular que corta al círculo |w| = ρ en w_{0} = μ (z_{0}), con el mismo ángulo α. Aplicando el teorema de Abel-Picard (véanse [2] y el Apéndice G) se tiene entonces que

∑_{k=0}^{∞}p_{k} · μ (z)^{k} = ∑_{k=0}^{∞}p_{k} · w^{k} ⟶ ∑_{k=0}^{∞}p_{k} · w_{0}^{k} = ∑_{k=0}^{∞}p_{k} · μ (z_{0})^{k} .  (2.15)

cuando w → w_{0}. Por otra parte, como la serie ∑_{k=0}^{∞} p_{k} · λ (z)^{k} converge uniformemente en ℂ, entonces

∑_{k=0}^{∞}p_{k} · λ (z)^{k} ⟶ ∑_{k=0}^{∞}p_{k} · λ (z_{0})^{k} ,  (2.16)

cuando z → z_{0}, y (2.14) se obtiene inmediatamente de (2.5), (2.15) y (2.16). □ ✓

Nota 2.1. Para información adicional sobre las representaciones conformes, el lector puede consultar [4].

2.3. Otras sucesiones de polinomios generadas por fórmulas lineales de recurrencia de segundo orden.

Considérese la fórmula de recurrencia

S_{n+1} − (a_{n} · z + b_{n}) · S_{n} + c_{n} · S_{n−1} = 0, n = 1, 2, 3, …  (2.17)

donde

a_{n} → a ≠ 0, b_{n} → b, c_{n} → c.

Mediante el cambio de variables

S_{n} = (\sqrt{c})^{n} · S̃_{n}, n = 0, 1, 2, …,

la fórmula (2.17) se transforma en

S̃_{n+1} − (\frac{1}{\sqrt{c}} · a_{n} · z + \frac{1}{\sqrt{c}} · b_{n}) · S̃_{n} + \frac{c_{n}}{c} · S̃_{n−1} = 0, n = 1, 2, 3, …,

donde

\frac{1}{\sqrt{c}} · a_{n} → \frac{1}{\sqrt{c}} · a, \frac{1}{\sqrt{c}} · b_{n} → \frac{1}{\sqrt{c}} · b, \frac{c_{n}}{c} → 1.

Podemos suponer entonces, sin pérdida de generalidad, que c = 1.

A su vez, mediante el cambio de parámetro

z = (\frac{2}{a}) · z′ − \frac{b}{a},

(2.17) se transforma en

S_{n+1} − (\frac{2a_{n}}{a} · z′ + b_{n} − \frac{a_{n}.b}{a}) · S_{n} + c_{n} · S_{n−1} = 0, n = 1, 2, 3, …,

donde

\frac{2a_{n}}{a} → 2, b_{n} − \frac{a_{n}b}{a} → 0.

Por esta razón podemos suponer también, sin pérdida de generalidad, que a_{n} → 2, b_{n} → 0, c_{n} → 1.

Nota 2.2. En la fórmula de recurrencia (2.17), la variable independiente es “n”, la variable dependiente es “S_{n}” y z es un “parámetro”; esto es, para cada valor de z tenemos una fórmula de recurrencia diferente.

Examinaremos ahora algunas propiedades básicas de la sucesión (S_{n} (z)) definida por (2.17). Para cada valor del parámetro z, consideremos nuevamente la fórmula de recurrencia

S_{n+1} (z) − (a_{n} · z + b_{n}) · S_{n} (z) + c_{n} · S_{n−1} (z) = 0, n = 1, 2, 3, …

donde ahora suponemos que a_{n} → 2, b_{n} → 0, c_{n} → 1. Supondremos además que a_{n}, b_{n,}c_{n} son reales y que satisfacen las siguientes condiciones:

a_{n} > |b_{n}| , c_{n} > 0, n ⩾ 1  (2.18)

y

∑_{n=1}^{∞} |b_{n}| < +∞, ∑_{n=1}^{∞} |a_{n} · a_{n−1} − 4 · c_{n}| < +∞.  (2.19)

Si la sucesión (S_{n} (z)) satisface las condiciones iniciales S_{0} (z) = 1, S_{1} (z) = z, entonces:

(i) S_{n} (z) es un polinomio de grado n en el parámetro z. Esto es evidente. (ii) Existe M ⩾ 1 tal que para x real, x > M, la sucesión (S_{n} (x)) es creciente y lim_{n→∞} S_{n} (x) = +∞ (exponencialmente).

En efecto, existe M ⩾ 1 tal que a_{n}·M > |b_{n}|+c_{n}+2 para todo n. Supóngase ahora, inductivamente, que 0 < S_{0} (x) < S_{1}(x) < ⋯ < S_{n}(x) para x > M. Entonces,

S_{n+1} (x) > (a_{n}M + b_{n}) · S_{n} (x) − c_{n} · S_{n−1} (x)
> (c_{n} + 2) · S_{n} (x) − c_{n} · S_{n−1} (x)
> 2.S_{n} (x) + c_{n} · (S_{n} (x) − S_{n−1} (x))
> 2 · S_{n} (x) .

Por lo tanto,

S_{n+1} (x) > S_{n} (x) y S_{n} (x) > 2^{n} · S_{0} (x) , n ⩾ 0.

Esto demuestra que (S_{n}(x)) es creciente y que lim_{n→∞} S_{n} (x) = +∞ (exponencialmente) para x > M.

(iii) Para x real, x < −M, la sucesión ((−1)^{n} · S_{n}(x); n = 0, 1, 2, …) es creciente y lim_{n→∞} (−1)^{n} · S_{n} (x) = +∞ (exponencialmente).

En efecto, mediante el cambio de variables S_{n} (x) = (−1)^{n}·S̃_{n} (x) , la fórmula de recurrencia (2.17) se transforma en

S̃_{n+1} (x) − (a_{n} · (−x) − b_{n}) · S̃_{n} (x) + c_{n} · S̃_{n−1} (x) = 0,

y basta utilizar el mismo argumento que en (ii).

(iv) Los ceros del polinomio S_{n} (x) son todos reales y están en el intervalo (−M, M).

En efecto, utilizando la fórmula de recurrencia (2.17), las propiedades (ii) y (iii), y el teorema del valor intermedio, se puede demostrar, por inducción, que siempre existe un cero de S_{n} (x) entre dos ceros consecutivos de S_{n−1}(x), que existe un cero de S_{n}(x) entre −M y el primer cero de S_{n−1}(x), y que hay también un cero de S_{n}(x) entre el último cero de S_{n−1}(x) y M.

Examinemos ahora el comportamiento asintótico de (S_{n} (x)) para z ∈ Ω. Efectuando el cambio de variables

S_{n} (z) = ∏_{k=1}^{n−1} \frac{a_{k}z + b_{k}}{2z} · X_{n} (z) , n ⩾ 2, S_{0} (z) = X_{0} (z) , S_{1} (z) = X_{1} (z) , (2.20)

la fórmula de recurrencia (2.17) toma la forma

X_{n+1} (x) − 2z · X_{n} (z) + c̃_{n} (z) · X_{n−1} (z) = 0, X_{0} (z) = 1, X_{1} (z) = z, (2.21)

donde

c̃_{n} (z) = c_{n} · \frac{4z^{2}}{(a_{n}z + b_{n}) (a_{n−1}z + b_{n−1})}, c̃_{1} (z) = c_{1} · \frac{2z}{a_{1}z + b_{1}} . (2.22)

De esto se deducen inmediatamente las siguientes propiedades:

(v) c̃_{n} (z) es una función análitica de z en la región Ω = ℂ ∖ [−1, 1]. Además, (c̃_{n} (z)) → 1 uniformemente en compactos de Ω.

(vi) X_{n} (z)es una función análitica de z en la región Ω.

(vii) La sucesión (∏_{k=1}^{n−1} \frac{|a_{k}z+b_{k}|}{2|z|}; n = 2, 3, 4, …) es moderada para z ∈ Ω. Esto es (véase [7]),

lim_{n→∞} \sqrt[n]{∏_{k=1}^{n−1} \frac{|a_{k}z + b_{k}|}{2 |z|}} = 1.

(viii)

∑_{n=1}^{∞} |c̃_{n} (z) − 1| < +∞.  (2.23)

En efecto,

|c̃_{n} (z) − 1| = \frac{|(a_{n} · a_{n−1} − 4c_{n}) · z^{2} + (a_{n}b_{n−1} + a_{n−1}b_{n}) · z + b_{n} · b_{n−1}|}{|a_{n}z + b_{n}| · |a_{n−1}z + b_{n−1}|},

y por la hipótesis (2.19), la serie (2.23) converge. Nótese que la suma de la serie (2.23) es uniformemente acotada sobre cualquier subconjunto compacto de Ω.

Sea α_{n} (z) definida por

α_{n} (z) = \frac{X_{n} (z)}{μ (z)^{n}}, n ⩾ 1.

Entonces α_{n} (z) es, para todo n, analítica en la región Ω. Ahora, por el Corolario 1.2, la sucesión (α_{n} (z) ; n = 0, 1, 2, …) converge para todo z ∈ Ω, y por (1.22), la sucesión (α_{n} (z)) es uniformemente acotada en cualquier subconjunto compacto de Ω. Por lo tanto, la sucesión (α_{n} (z)) converge uniformente en compactos de Ω (véase [5]).

Sea

α (z) = lim_{n→∞}α_{n} (z) , z ∈ Ω.

Entonces α (z) es analítica en Ω. Como para todo n, α_{n} (z) ≠ 0 para todo z ∈ ℂ ∖ [−M, M] ⊆ Ω (Propiedad (iv)), entonces α (z) no tiene ningún cero en ℂ ∖ [−M, M], o a(z) = 0 para todo z ∈ ℂ ∖ [−M, M] (Teorema de Hurwitz, véase [4] p. 247). Supóngase que fuera α (z) = 0 para todo z ∈ ℂ ∖ [−M, M]. Por el Teorema de coincidencia de las funciones analíticas ([4], p. 226), se tendría que α (z) = 0 para todo z ∈ Ω y, por el Corolario 1.2, lim_{n→∞} X_{n}(z)/λ(z)^{n} existiría para todo z ∈ Ω. Por lo tanto,

lim_{n→∞}X_{n} (z) = lim_{n→∞} \frac{X_{n} (z)}{λ (z)^{n}} · λ (z)^{n} = 0, z ∈ Ω.

Esto contradice la propiedad (ii). Entonces (ix)

lim_{n→∞}α_{n} (z) = lim_{n→∞} \frac{X_{n} (z)}{μ (z)^{n}} = α (z)

uniformemente en compactos de Ω , donde α(z) es análitica en Ω y α(z) ≠ 0 para todo z en ℂ ∖ [−M, M] ⊆ Ω.

De (2.20 ) se obtiene entonces que (S_{n} (z)) se comporta asintóticamente en la forma

S_{n} (z) ∼ α (z) · ∏_{k=1}^{n−1} \frac{a_{k}z + b_{k}}{2z} · μ (z)^{n} ,  (2.25)

de lo cual (x)

lim_{n→∞} \frac{S_{n} (z)}{S_{n−1} (z)} = μ (z) , z ∈ ℂ ∖ [M, M] ·  (2.26)

Nótese que la función análitica α (z) puede anularse en puntos del intervalo [−M, M] (véase el siguiente ejemplo), y si α (x_{0}) = 0 entonces, por el Corolario 1.2,

lim_{n→∞} \frac{S_{n} (x_{0})}{S_{n−1} (x_{0})} = λ (x_{0})

de lo cual, lim_{n→∞} \sqrt[n]{|S_{n} (x_{0})|} = |λ (x_{0})|.

Ejemplo 2.1. Sea (S_{n} (x) ; n = 0, 1, 2, …) la solución de la fórmula lineal de recurrencia

S_{2} (z) − (z − 1) · S_{1} (z) + S_{0} (z) = 0,
S_{n+1} (z) − 2z · S_{n} (z) + S_{n−1} (z) = 0, n ≥ 2,

sujeta a las condiciones iniciales S_{0} (z) = 1, S_{1} (z) = z.

Entonces,

S_{2} (z) = z^{2} − z − 1; S_{n} (z) = A (z) · μ (z)^{n} + B (z) · λ (z)^{n} , n ⩾ 1,

y A (z) , B (z) pueden determinarse a partir de las condiciones S_{1} (z) = z, S_{2} (z) = z^{2} − z − 1; o sea,

A (z) · μ (z) + B (z) · λ (z) = z, A (z) · μ (z)^{2} + B (z) · λ (z)^{2} = z^{2} − z − 1.

Entonces,

A (z) = \frac{z^{2} − z − 1 − z · λ (z)}{μ (z) · (μ (z) − λ (z))}, B (z) = \frac{z · μ (z) − z^{2} + z + 1}{λ (z) · (μ (z) − λ (z))}·

Como además S_{n}(z)/μ(z)^{n} ∼ A (z) y

\frac{S_{n} (z)}{μ (z)^{n}} = \frac{z − 1}{2z} · α_{n} (z) ∼ \frac{z − 1}{2z} · α (z)

entonces α (z) = 2z/(z − 1) · A (z) para z ∈ Ω. Como λ (x) = x + \sqrt{x^{2} − 1} para x < −1, se ve inmediatamente que α (x) no tiene ceros en (−∞, −1) . Por otra parte, λ (x) = x − \sqrt{x^{2} − 1}, x > 1, y α (x) = 0 si y sólo si x^{2} − x − 1 − x · (x − \sqrt{x^{2} − 1}) = 0; o sea, cuando x > 1, α (x) = 0 si y sólo si x^{3} − x^{2} − x − 1 = 0. Ahora, la ecuación cúbica x^{3} − x^{2} − x − 1 = 0 tiene la única raíz real q = 1.8393 … > 1.

Por lo tanto,

S_{n} (z) ∼ \frac{z − 1}{2z} · α (z) · μ (z)^{n} , z ∈ Ω,

donde α (z) es analítica en Ω, α (z) ≠ 0 para z ≠ q y α (q) = 0. Además, existe una constante real C ≠ 0 tal que S_{n} (q) ∼ C · λ (q)^{n} . En consecuencia,

lim_{n→∞} \frac{S_{n} (q)}{S_{n−1} (q)} = λ (q) ·

Nota 2.3. Sea Z el conjunto de los ceros de todos los polinomios S_{n} (x) , n = 1, 2, 3, . . .. Es decir,

Z = ⋃_{n=1}^{∞}{x : S_{n} (x) = 0}.  (2.27)

Si x_{0} ∉ [−1, 1] es un punto de acumulación del conjunto Z entonces α (x_{0}) = 0, ya que (α_{n} (x) ; n = 0, 1, 2, …) converge a α (x) uniformemente en compactos de Ω.

(xi) La sucesión de funciones (α_{n} (z) ; n = 0, 1, 2, …) es de variación uniformemente acotada en cualquier subconjunto compacto de Ω (Apéndice E).

En efecto, reemplazando X_{n} (z) = α_{n} (z) · μ (z)^{n} en (2.21) se obtiene la fórmula de recurrencia

α_{n+1} (z) − \frac{2z}{μ (z)} · α_{n} (z) + \frac{c̃_{n} (z)}{μ (z)^{2}} · α_{n−1} (z) = 0.

Teniendo entonces en cuenta que 2z = λ (z) + μ (z) y 1 = λ (z) · μ (z) , la fórmula anterior puede escribirse en la forma

α_{n+1} (z) − (1 + \frac{λ (z)}{μ (z)}) · α_{n} (z) + \frac{λ (z)}{μ (z)} · c̃_{n} (z) · α_{n−1} (z) = 0, n = 1, 2, 3, …,

o sea, en la forma

α_{n+1} (z) − α_{n} (z) = \frac{λ (z)}{μ (z)} · (α_{n} (z) − α_{n−1} (z)) + \frac{λ (z)}{μ (z)} · (1 − c̃_{n} (z)) · α_{n−1} (z) . (2.28)

Si D es un subconjunto compacto Ω, existen 0 < r < 1 y M > 0 tales que, para todo z ∈ D y todo n ⩾ 1,

|\frac{λ (z)}{μ (z)}| < r, |α_{n−1} (z)| < \frac{M}{r}

En efecto, |\frac{λ(z)}{μ(z)} | < 1 para todo z ∈ D, y la sucesión (α_{n−1} (z)) es uniformemente acotada en D. De (2.28) se obtiene además la desigualdad

|α_{n+1} (z) − α_{n} (z)| ⩽ r · |α_{n} (z) − α_{n−1} (z)| + M · |1 − c̃_{n} (z)| ,

así que sumando con respecto a n, desde n = 1 hasta n = N,

(1 − r) · ∑_{n=2}^{N} |α_{n} (z) − α_{n−1} (z)| ⩽ r · |α_{1} (z) − α_{0} (z)| + M · ∑_{n=1}^{N} |1 − c̃_{n} (z)| , (2.29)

y (2.29) implica que la sucesión (α_{n} (z)) es de variación uniformemente acotada en D.

(xii) Se tiene que

∏_{k=1}^{n−1} \frac{a_{k}z + b_{k}}{2z} = ∏_{k=1}^{n−1} \frac{a_{k}}{2} · ∏_{k=1}^{n−1} (1 + \frac{b_{k}}{a_{k}z} ) ,

y la sucesión ( ∏_{k=1}^{n−1}(1 + \frac{b_{k}}{a_{k}z})) es de variación uniformemente acotada en cual-

quier subconjunto compacto Ω.

En efecto,

∑_{n=1}^{∞} |∏_{k=1}^{n}(1 + \frac{b_{k}}{a_{k}z}) − ∏_{k=1}^{n−1}(1 + \frac{b_{k}}{a_{k}z})| = ∑_{n=1}^{∞} |\frac{b_{n}}{a_{n}z} | · |∏_{k=1}^{n−1}(1 + \frac{b_{k}}{a_{k}z})| ,

y, por (2.19) ∑_{n=1}^{∞} |b_{n}| < +∞. Entonces, la serie anterior converge, y su suma total, es acotada en cualquier subconjunto compacto de Ω.

Consideremos ahora la serie

∑_{n=0}^{∞}p_{n} · S_{n} (z) , lim sup \sqrt[n]{|p_{n}|} = \frac{1}{ρ} < 1.  (2.30)

Esta serie converge en el interior de la elipse en (2.13). En caso de que la función analítica α (z) tenga un cero x_{0} en el exterior de tal elipse, la serie converge en x_{0}, pues lim_{n→∞} \sqrt[n]{|S_{n}(x_{0})|} = |λ(x_{0})| < 1. Con la posible excepción en los ceros de α (z) , la serie diverge en el exterior de la elipse (2.13). Si todos los ceros de los polinomios S_{n} (z) , n = 0, 1, 2, … están en el intervalo (−1, 1) entonces α (z) ≠ 0 en la región Ω, y en tal caso la serie (2.30) diverge sin excepciones en el exterior de la elipse (2.13).

Teorema 2.2. (Teorema del límite de Abel-Picard). Sea z_{0} un punto sobre la elipse (2.13), así que (|μ (z_{0})|) = ρ. Si α (z_{0}) ≠ 0 y la serie (2.30) converge en z_{0}, entonces

∑_{n=0}^{∞}p_{n} · S_{n} (z) ⟶∑_{n=0}^{∞}p_{n} · S_{n} (z_{0})  (2.31)

cuando z, con |μ (z)| < ρ, tiende a z_{0} sobre cualquier curva regular que corte a la elipse (2.13) en z_{0}, y que forme con ésta un ángulo de corte no nulo.

Demostración. Sea D un disco cerrado de Ω, con centro en z_{0} (Figura 3).

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.................................................................................................................................................................................................................................................... ........................................................................................................................................................................................................................................................  x y  • • −1  1 0  \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 a  b K  z z_{0} D ↖ [[[  ]]]

Figura 3

Teniendo en cuenta que

\frac{S_{n} (z)}{μ (z)^{n}} = ∏_{k=1}^{n−1} \frac{a_{k}}{2} · ∏_{k=1}^{n−1}(1 + \frac{b_{k}}{a_{k}z}) · α_{n} (z) ,

la sucesión

(\frac{S_{n} (z)}{S_{n} (z_{0})} (\frac{μ (z_{0})}{μ (z)})^{n}; n = 2, 3, …) = (\frac{α_{n} (z) · ∏_{k=1}^{n−1}(1 + \frac{b_{k}}{a_{k}z})}{α_{n} (z_{0}) · ∏_{k=1}^{n−1}(1 + \frac{b_{k}}{a_{k}z_{0}} )}; n = 2, 3, …)

resulta ser de variación uniformemente acotada en D, pues las sucesiones

(α_{n} (z)) y ( ∏_{k=1}^{n−1} (1 + \frac{b_{k}}{a_{k}·z} )) son de variación uniformemente acotada en D

y están separadas de cero. Por el lema de Abel ([2] y Apéndice F), la serie

∑_{n=0}^{∞} (p_{n} · \frac{S_{n} (z)}{μ (z)^{n}}) · μ (z_{0})^{n} = ∑_{n=0}^{∞}p_{n} · S_{n} (z_{0}) · \frac{S_{n} (z)}{S_{n} (z_{0})} · (\frac{μ (z_{0})}{μ (z)})^{n}

converge uniformemente en D, pues la serie ∑_{n=0}^{∞} p_{n} · S_{n}(z_{0}), siendo independiente de z, converge uniformemente en D. Por medio de la misma representación conforme w = μ (z) utilizada anteriormente, y de la generalización del teorema del límite de Abel-Picard (Apéndice G), se obtiene entonces que

∑_{n=0}^{∞}p_{n} · S_{n} (z) =

∑_{n=0}^{∞}p_{n} · \frac{S_{n} (z)}{μ (z)^{n}} · μ (z)^{n} ⟶_{z→z_{0}} ∑_{n=0}^{∞}p_{n} · \frac{S_{n} (z_{0})}{μ (z_{0})^{n}} · μ (z_{0})^{n}

= ∑_{n=0}^{∞}p_{n} · S_{n} (z_{0}) ,

lo cual completa la demostración.

Nota 2.4. Si α (z_{0}) = 0, la serie (2.30) converge en z_{0}, pero en este caso no es válido el teorema de Abel.

Examinaremos finalmente el comportamiento de la serie (2.30) en ciertos casos particulares especialmente notables, relacionados con los denominados polinomios ortogonales clásicos.

Como en la fórmula de recurrencia (2.17) hemos supuesto que a_{n} → 2, b_{n} → 0, c_{n} → 1, los casos más simples corresponden a

a_{n} = 2 + \frac{A}{n} + O (\frac{1}{n^{2}} ) , b_{n} = \frac{A}{n}′ + \frac{B}{n^{2}} + O (\frac{1}{n^{3}} ) ,

c_{n} = 1 + \frac{C}{n} + O (\frac{1}{n^{2}} ) .

Tenemos entonces que

a_{n} · a_{n−1} − 4 · c_{n} = \frac{4 (A − C)}{n} + O (\frac{1}{n^{2}} ) .

Por lo tanto, (2.18) y (2.19) se satisfacen si y sólo si A′ = 0, A = C; esto es,

\begin{aligned}a_{n} = 2 + \frac{A}{n} & + O (\frac{1}{n^{2}} ) , b_{n} = \frac{B}{n^{2}} + O (\frac{1}{n^{3}} ) , \\ & c_{n} = 1 + \frac{A}{n} + O (\frac{1}{n^{2}} ) . \end{aligned}  (2.32)

Distinguiremos cuatro casos:

1. A = B = 0. Entonces a_{n} = 2, b_{n} = 0, c_{n} = 1. Se obtiene así la fórmula de recurrencia de los polinomios de Chebyshev (T_{n} (z) ; n = 0, 1, 2, …) :

T_{n+1} (z) − 2z · T_{n} (z) + T_{n−1} (z) = 0, n ⩾ 1.

2. A = −1, B = 0. Corriendo el subíndice n en 1 se obtiene que

a_{n} = 2 − \frac{1}{n + 1}, b_{n} = 0, c_{n} = 1 − \frac{1}{n + 1}.

En este caso la fórmula de recurrencia (2.17) genera (véase [4]) los polinomios de Legendre (P_{n} (z) ; n = 0, 1, 2, …) :

P_{n+1} (z) − \frac{2n + 1}{n + 1} · z · P_{n} (z) + \frac{n}{n + 1} · P_{n−1} (z) = 0, n ⩾ 1.

3. B = 0, A = 2 (ν − 1). Reemplazando n por n + 1 se obtiene que

a_{n} = 2 + \frac{2 (ν − 1)}{n + 1} = \frac{2n + 2ν}{n + 1} , b_{n} = 0,

c_{n} = 1 + \frac{2(ν − 1)}{n + 1} = \frac{n + 2ν − 1}{n + 1},

lo cual reduce (2.17) a la fórmula de recurrencia (véase [4]) de los polinomios de Gegenbauer (G_{n}^{ν} (z) , n = 0, 1, 2, …) :

G_{n+1}^{ν} (z) − \frac{2n + 2ν}{n + 1} · z · G_{n}^{ν} (z) + \frac{n + 2ν − 1}{n + 1} · G_{n−1}^{ν} (z) = 0, n ⩾ 1.

4. A = −1, B ≠ 0. Sean, por ejemplo,

\begin{matrix}a_{n} = \frac{(α + β + 2n + 1) (α + β + 2n + 2)}{2 (n + 1) (α + β + n + 1)} ⟶ 2, \\ b_{n} = \frac{(α^{2} − β^{2}) (α + β + 2n + 1)}{2 (n + 1) (α + β + 2n) (α + β + n + 1)} ⟶ 0, \\ c_{n} = \begin{matrix}(α + n) & (β + n) & (α + β + 2n + 2) \\ (n + 1) & (α + β + n + 1) & (α + β + 2n) \end{matrix} ⟶ 1, \end{matrix}  (2.33)

Donde α, β son constantes, α > −1, β > −1.

La fórmula de recurrencia (2.17) genera en este caso los denominados polinomios de Jacobi (J_{n} (α, β; z) ; n = 0, 1, 2, …) (véase [4]). Se observa inmediatamente que

\begin{aligned}a_{n} = 2 − \frac{1}{n} & + O (\frac{1}{n^{2}} ) , b_{n} = \frac{α^{2} − β^{2}}{2n^{2}} + O (\frac{1}{n^{3}} ) , \\ & c_{n} = 1 − \frac{1}{n} + O (\frac{1}{n^{2}} ) , \end{aligned}

así que los coeficientes a_{n}, b_{n}, c_{n} en (2.33) satisfacen (2.32) con

A = −1, B = \frac{α^{2} − β^{2}}{2} .

Los polinomios de los cuatro casos anteriores se conocen como los polinomios “ortogonales” clásicos. Los ceros de todos estos polinomios están en (−1, 1).

Por lo tanto, son válidos para ellos todos los resultados obtenidos en el presente parágrafo: en particular, el teorema del límite de Abel, sin puntos excepcionales.

Apéndices

A. Relaciones de recurrencia de primer orden.

Sea (X_{n}) una solución de la fórmula lineal de recurrencia de primer orden

X_{n+1} = A_{n} · X_{n} + B_{n}, A_{n} → A, |A| < 1.

Si ∑_{n=1}^{∞} |B_{n}| < +∞, entonces ∑_{n=1}^{∞} |X_{n}| < +∞ (véase [12]). En efecto, podemos suponer, sin pérdida de generalidad, que existe r < 1 tal que |A_{n}| < r para todo n. Entonces

|X_{n+1}| < r · |X_{n}| + |B_{n}| , n = 1, 2, 3, …

Sumando la desigualdad anterior desde n = 1 hasta n = N se obtiene que

∑_{n=2}^{N+1} |X_{n}| ⩽ r · ∑_{n=1}^{N} |X_{n}| + ∑_{n=1}^{N} |B_{n}| ,

o sea, que

(1 − r) · ∑_{n=2}^{N} |X_{n}| ⩽ |X_{1}| + ∑_{n=1}^{N} |B_{n}| .

Esto implica que ∑_{n=1}^{∞} |X_{n}| < +∞.

Un argumento similar demuestra que si (X_{n}) es la solución “convergente” de la fórmula anterior cuando A_{n} → A, |A| > 1, entonces ∑_{n=1}^{∞} |B_{n}| < ∞ implica que ∑_{n=1}^{∞} |X_{n}| < +∞ (véase [12]).

B. El método de variación de parámetros.

Es bien conocido que la fórmula lineal de recurrencia de segundo orden con coeficientes constantes,

X_{n+1} − a · X_{n} + c · X_{n−1} = 0, n = 1, 2, 3, …,

tiene dos soluciones (λ^{n}; n = 0, 1, 2, …) , (μ^{n}; n = 0, 1, 2, …), donde λ, μ son las raíces de la ecuación cuadrática t^{2} − a · t + c = 0. Consideremos ahora la fórmula de recurrencia no-homogénea

X_{n+1} − a · X_{n} + c · X_{n−1} = e_{n}, n = 1, 2, 3, …  (1)

Como λ + μ = a, λ · μ = c, la fórmula (1) puede escribirse en la forma

(X_{n+1} − λX_{n}) = μ · (X_{n} − λX_{n−1}) + e_{n}, n = 1, 2, 3, …,

Esta es una fórmula lineal de recurrencia de primer orden para la sucesión (X_{n} − λX_{n−1}) . Su solución general es (véase [6]).

(X_{n+1} − λX_{n}) = μ^{n} · [(X_{1} − λX_{0}) + ∑_{k=1}^{n} \frac{e_{k}}{μ^{k}} ] , n ⩾ 1.

Intercambiando λ y μ se obtiene que

(X_{n+1} − μX_{n}) = λ^{n} [(X_{1} − μX_{0}) + ∑_{k=1}^{n} \frac{e_{k}}{λ^{k}} ] .

Sustrayendo una de otra las igualdades anteriores, se obtinene que la solución general (X_{n}) de (1) está dada por

X_{n} = \frac{μ^{n} − λ^{n}}{μ − λ} X_{1} − \frac{λμ (μ^{n−1} − λ^{n−1})}{μ − λ}X_{0} + \frac{1}{μ − λ} · ∑_{k=1}^{n−1} (μ^{n−k} − λ^{n−k}) · e_{k}. (2)

En forma similar se obtiene que la solución general de la fórmula (1) cuando λ = μ (a^{2} − 4c = 0) es

X_{n} = μ^{n−1} · [nX_{1} − (n − 1) μ · X_{0}] + ∑_{k=1}^{n−1} (n − k) · μ^{n−k−1} · e_{k}.  (3)

C. Acotación de las soluciones.

Supóngase que la sucesión (X_{n}) satisface para todo n ≥ 1 la desigualdad (véase [12])

|X_{n}| ⩽ M + ∑_{k=0}^{n−1}d_{k} · |X_{k}|

donde d_{k} ⩾ 0 para todo k ⩾ 0 y |X_{0}| ⩽ M. Entonces,

|X_{n}| ⩽ M · ∏_{k=0}^{n−1} (1 + d_{k}) ,  (4)

para todo n ⩾ 1.

En efecto, razonando por inducción se tiene en primer lugar que

X_{1} ⩽ M + d_{0} · |X_{0}| ⩽ M · (1 + d_{0}) .

Supóngase ahora que (4) vale para n. Entonces,

|X_{n+1}| ⩽ M + d_{0} |X_{0}| + ∑_{k=1}^{n}d_{k} · M · (1 + d_{0}) (1 + d_{1}) ⋯ (1 + d_{k−1})

= M + M · d_{0} + M · ∑_{k=1}^{n}[(1 + d_{0}) ⋯ (1 + d_{k}) − (1 + d_{0}) ⋯ (1 + d_{k−1})]

= M + M · d_{0} + M · [(1 + d_{0}) ⋯ (1 + d_{n}) − (1 + d_{0})]

= M · ∏_{k=0}^{n} (1 + d_{k}) ,

lo cual demuestra la afirmación.

D. Sucesiones de variación acotada.

Se dice que una sucesión (B_{k}; k = 0, 1, 2, …) es de variación acotada, si

∑_{k=0}^{∞} |B_{k} − B_{k+1}| < +∞.

Toda sucesión de variación acotada converge (véanse [2], [12]).

E. Sucesiones de variación uniformemente acotada.

Se dice que una sucesión (B_{k} (t) ; k = 0, 1, 2, …) de funciones definidas en un dominio T es de variación uniformemente acota en T si es uniformemente acotada en T y existe una constante M > 0 tal que ∑_{k=0}^{∞} |B_{k} (t) − B_{k+1} (t)| ⩽ M para todo t ∈ T (véase [2]).

F. Un lema de Abel.

El siguiente resultado de Abel es importante para nuestros propoósitos.

Sea ∑_{k=0}^{∞} B_{k}(t) una serie de funciones uniformemente convergente en T. Si la sucesión de funciones (A_{k} (t) ; k = 0, 1, 2, …) es de variación uniformemente acotada en T, entonces la serie ∑_{k=0}^{∞} A_{k} (t) · B_{k} (t) converge uniformente en T.

La demostración resulta inmediatamente de la condición de convergencia uniforme de Cauchy.

G. Un teorema de Abel-Picard.

La siguiente es la forma del teorema de Abel-Picard a la que recurriremos. Sea (A_{n} (z) ; n = 0, 1, 2, …) una sucesión de funciones continuas tal que

lim sup \sqrt[n]{|A_{n} (z)|} = \frac{1}{ρ}

independientemente de z. Entonces, |z| = ρ es el círculo de convergencia de la serie ∑_{n=0}^{∞} A_{n} (z) · z^{n}, y si z_{0} es un punto sobre el círculo de convergencia y la serie ∑_{n=0}^{∞} A_{n} (z)·z_{0}^{n} converge uniformemente en una vecindad de z_{0}, entonces

∑_{n=0}^{∞}A_{n} (z) · z^{n} ⟶ ∑_{n=0}^{∞}A_{n} (z_{0}) · z_{0}^{n},  (5)

cuando z (con |z| < ρ) tiende a z_{0} a lo largo de cualquier curva regular que corte al círculo |z| = ρ en z_{0} formando con éste un ángulo no nulo. Demostración. Sea K una curva regular que corta el círculo |z| = ρ en z_{0}, formando con éste un ángulo φ no nulo (Figura 4).

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..............................................................................................................................................................................................................................................................................................................  x y  • • 0  ^{(((} K  z  z_{o} ρ φ

Figura 4

Teniendo en cuenta que la sucesión ((\frac{z}{z_{0}} )^{n}; n = 0, 1, 2, …) es de variación acotada uniformemente en K (véase [2]), el lema de Abel en el Apéndice F asegura que la serie ∑_{n=0}^{∞} A_{n} (z) · z^{n} = ∑_{n=0}^{∞} A_{n} (z) · z_{o}^{n} · (\frac{z}{z_{0}} )^{n} converge uniformemente en una vecindad de z_{0} en K. Esto implica la validez de (5).

Como un caso particular de lo anterior, si A_{n} (z) = A_{n} es independiente de z, se obtiene el teorema usual de Abel-Picard, el cual, si la curva K es el radio que toca z_{0}, se conoce simplemente como el teorema del límite de Abel (véase [1]).

Agradeciemientos. El autor agradece al revisor del artículo su valiosa ayuda en la preparación de la versión final.

Referencias

[1] Apostol, T. M., Mathematical Analysis, Addison-Wesley, 1963.

[2] Bromwich, T. J., An Introduction to the Theory of Infinite Series, Macmillan, 1926.

[3] Dumett, M., Productos de Hadamard de series de polinomios ortogonales clásicos, Lect. Mat., 16 no. 1 (1995), 37–62.

[4] Erdelyi, Magnus, Oberhettinger, Tricomi, Higher Transcendental Fuctions, Vol.II, Cap.X, Mc Graw-Hill, 1953.

[5] Rudin, W., Real and Complex Analysis, Mc-Graw Hill, 1966.

[6] Takeuchi, Y., Estudio sistemático de algunas sucesiones, Mat. Ens. Univ., 20 (1981), 3–74.

[7] Takeuchi, Y., Sucesiones moderadas, Bol. Mat. (Nueva serie) 1 no. 1 (1994), 17–34.

[8] Takeuchi, Y., Regla de L’Hóspital para series, Bol. Mat. (Nueva serie), 1 no. 2 (1994), 25-38.

[9] Takeuchi, Y., Convergencia de sucesiones dadas por fórmulas de recurrencia de primer orden, Rev. Colombiana Mat. 27 (1993), 111–125.

[10] Takeuchi, Y., Conjunto de convergencia y conjunto ternario de Cantor, Bol. Mat. (Nueva serie), 2 (1995), 141–165.

[11] Takeuchi, Y., Algunas observaciones sobre la convergencia de los productos infinitos, próximo a publicarse en Lect. Mat.

[12] Takeuchi Y., Notas del Seminario de Sucesiones y Fórmulas de Recurrencia, Departamento de Matemáticas y Estadística, Universidad Nacional de Colombia, Bogotá, 1994.

(Recibido en junio de 1998; revisado por el autor en noviembre de 1998)

Departamento de Matemáticas y Estadística

Universidad Nacional de Colombia Bogotá, Colombia