Use the defining equation for Bessel functions which explicitly contains the second derivative as well as other terms,since Bessel functions are defined

as eigenfunctions of Laplace operator in cylindrical coordinates.The functional identities you mention are consequences of this equation.

There are various ways to write the second derivative of the Bessel function in terms of higher and lower orders of Bessel functions.

For instance using the fact that

J'n(x)=1/2(Jn-1(x)-Jn+1(x)) it is possible to show that

J''n+1(x)=1/4(Jn-1(x)-2Jn+1(x)+Jn+3(x)).

Of course the series expression for Jm(x) is well known (see for instance http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html)

Using the series expression for Jm(x) you can easily find an explicit expression for J''n+1.

Thanks for all the answers

Hi Manuele

But I seek J"n+1 as a function of Jn and Jn+1

Sorry, for misreading your question. Assume the following identity (Whittaker and Watson "A course of modern analysis" page 359):

(1) Jm-1(x)-Jm+1(x)=(2m/x)Jm(x)

You can use this to rewrite Jn-1 Jn+3 in terms of Jn and Jn+1 in:

J''n+1(x)=1/4(Jn-1(x)-2Jn+1(x)+Jn+3(x)).

In fact, from (1):

Jn-1(x)=(2n/x)Jn(x)+Jn+1(x)

From (1) with m=n+2 :

Jn+3(x)=-[2(n+2)/x ]Jn+2(x)+Jn+1

But from (1) with m=n+1:

Jn+2(x)=-(2(n+1)/x)Jn+1(x)+Jn(x)

Similarly, you can use identity (1) to write most other expressions for Jn+1'' in terms of Jn and Jn+1.

Thanks Dear Manuele

Best wishes