Numero armonico

In matematica, per ogni intero naturale n si definisce come n-esimo numero armonico la somma:

H_{n}=1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}

Si tratta evidentemente di numeri razionali e si dimostra che le corrispondenti frazioni ridotte ai minimi termini hanno numeratore dispari e denominatore pari.

In concreto i primi termini della successione dei numeri armonici sono:

1, 3/2, 11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, 7381/2520, 83711/27720, ...

I numeratori dei numeri armonici sono detti numeri di Wostenholme e costituiscono la successione A001008 dell'OEIS. I denominatori costituiscono la successione A002805 dell'OEIS.

I numeri armonici costituiscono le somme parziali della serie armonica, notoriamente divergente.

Numeri armonici alternati

I numeri armonici sono strettamente collegati a quelli che si possono chiamare numeri armonici alternati

H'_{n}:=\sum _{k=1}^{n}(-1)^{k+1}{\frac {1}{k}}

Questi sono le somme troncate della serie armonica alternata notoriamente convergente e sono esprimibili mediante i numeri armonici dalle formule

H'_{2n}=H_{2n}-H_{n}\qquad H'_{2n+1}=H_{2n}-H_{n}+{\frac {1}{2n+1}}

Espressione analitica

I numeri armonici (e quindi anche i numeri armonici alternati) si possono esprimere analiticamente come

\,H_{n}=\gamma +\Psi (n+1)

mediante la costante di Eulero - Mascheroni e la funzione digamma (e di conseguenza mediante la funzione gamma)

\Psi (z)\equiv \psi _{0}(z):={\frac {d}{dz}}\ln \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}.

Numeri armonici generalizzati

Il concetto di numero armonico può essere generalizzato con la seguente definizione.

Fissati due interi naturali m ed n, si definisce come n-esimo numero armonico generalizzato di esponente m la somma:

H_{n}^{(m)}:=\sum _{k=1}^{n}{\frac {1}{k^{m}}}

Si può notare che i numeri armonici sono il caso particolare di numeri armonici generalizzati di esponente 1.

Per m negativi, si ottiene la somma di potenze di interi successivi

\sum _{k=1}^{n}k^{|m|}

che è strettamente legata ai polinomi di Bernoulli.

Collegamenti

I numeri armonici e i numeri armonici generalizzati che li comprendono si incontrano in numerose aree della matematica riferibili alla combinatoria e allo studio delle funzioni speciali. Essi intervengono nello studio di funzioni speciali particolari, ad es. della funzione poligamma della funzione polilogaritmo e della funzione zeta di Riemann; essi inoltre si incontrano in recenti sviluppi di elevata generalità, come le questioni collegate all'approssimazione di Hermite-Padé.

Bibliografia

R. L. Graham, D. E. Knuth and O. Patashnik (1990): Concrete Mathematics, Addison-Wesley, p. 259.
D. E. Knuth: The Art of Computer Programming. Addison-Wesley, Vol. 1, p. 615.

Voci correlate

Identità con numeri armonici

Altri progetti

Wikimedia Commons contiene immagini o altri file sul numero armonico

Collegamenti esterni

Numero armonico, in Enciclopedia della Matematica, Istituto dell'Enciclopedia Italiana, 2013.
(EN) harmonic number, su Enciclopedia Britannica, Encyclopædia Britannica, Inc.
(EN) Eric W. Weisstein, Harmonic Number, su MathWorld, Wolfram Research.
(EN) Harmonic number, su Encyclopaedia of Mathematics, Springer e European Mathematical Society.

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