Math Problems Vault

Problem 3.207

We use the same notation and hypothesis as before and expand

\frac{f(z)}{1-w\varphi '(z)} = \sum _{n=0}^{\infty } \frac{w^{n}}{n!}\left[\frac{d^{n}f(x)[\varphi (x)]^{n}}{dx^{n}}\right]_{x=0}.

Derive this formula from Lagrange's formula or Lagrange's formula from this one by correctly using the generality of both formulas. [The existence of one formula for a certain $f(z)$ implies immediately the other formula for another $f(z)$ .]

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Problem 3.208

Prove the formula in problem 207 directly by expressing the coefficient of $w^{n}$ as a Cauchy integral.

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Problem 3.209

Expand in ascending powers of $w$ the solution of the transcendental equation

ze^{-z} = w

that vanishes for $w=0$ .

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Problem 3.210

(continued). Expand $e^{\alpha z}$ in powers of $w=ze^{-z}$ where $\alpha$ is an arbitrary constant.

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Problem 3.211

Expand in ascending powers of $w$ the solution $x$ of the trinomial equation

1 - x + wx^{\beta } = 0

that becomes $1$ for $w=0$ .

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Problem 3.212

(continued). Expand $x^{\alpha }$ in powers of $w$ , where $\alpha$ denotes an arbitrary constant. ( $x^{\alpha }=y$ is the solution of the trinomial equation

1 - y^{1/\alpha } + wy^{\beta /\alpha } = 0.)

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Problem 3.213

(continued). Note the cases $\beta =0,1,2,-1,\frac{1}{2}$ and derive problems 209, 210 by taking the limit in problems 211, 212.

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Problem 3.214

Evaluate the sum of the power series

1 + \sum _{n=1}^{\infty } \frac{(n+\alpha )^{n}w^{n}}{n!}.

What is its radius of convergence?

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Problem 3.215

Prove problem 156 with the help of the results of problem 214.

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Problem 3.216

Let $\alpha$ and $\beta$ denote rational numbers. Then the series

1 + \binom {\alpha +\beta }{1}w + \binom {\alpha +2\beta }{2}w^{2} + \cdots + \binom {\alpha +n\beta }{n}w^{n} + \cdots

represents an algebraic function of $w$ .

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Problem 3.217

Arrange the successive powers of the trinomial $1+w+w^{2}$ in a regular triangular array

1

1 + w + w^{2}

1 + 2w + 3w^{2} + 2w^{3} + w^{4}

1 + 3w + 6w^{2} + 7w^{3} + 6w^{4} + 3w^{5} + w^{6}

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

The sum of the middle terms (in boldface: $1, w, 3w^{2}, 7w^{3}, \ldots$ ) is

1 + w + 3w^{2} + 7w^{3} + \cdots = \frac{1}{\sqrt{1-2w-3w^{2}}}.

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Problem 3.218

Arrange the successive powers of the binomial $1+w$ in a regular triangular array (Pascal's triangle)

1

1 + w

1 + 2w + w^{2}

1 + 3w + 3w^{2} + w^{3}

1 + 4w + 6w^{2} + 4w^{3} + w^{4}

\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots

Find the sum of the middle terms (in boldface) and, more generally, the sum of any column.

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Problem 3.219

(1)(2)(3)

Find the generating functions of the polynomials $P_{n}(x)$ , $P_{n}^{(\alpha ,\beta )}(x)$ , $L_{n}^{(\alpha )}(x)$ , defined by the formulas

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(1)

$P_{n}(x) = \dfrac {1}{2^{n}n!}\dfrac {d^{n}}{dx^{n}}(x^{2}-1)^{n}$ (Legendre's polynomials);

(2)

$(1-x)^{\alpha }(1+x)^{\beta }P_{n}^{(\alpha ,\beta )}(x) = \dfrac {(-1)^{n}}{2^{n}n!}\dfrac {d^{n}}{dx^{n}}(1-x)^{n+\alpha }(1+x)^{n+\beta }$ , $\alpha >-1$ , $\beta >-1$ (Jacobi's polynomials);

(3)

$e^{-x}x^{\alpha }L_{n}^{(\alpha )}(x) = \dfrac {1}{n!}\dfrac {d^{n}}{dx^{n}}e^{-x}x^{n+\alpha }$ , $\alpha >-1$ (generalized Laguerre's polynomials).

Remark.

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Cf.Part VI, problems 84, 98, 99. The generating function of the Legendre polynomials is the series

P_{0}(x) + P_{1}(x)w + P_{2}(x)w^{2} + \cdots + P_{n}(x)w^{n} + \cdots = 1 + xw + \frac{3x^{2}-1}{2}w^{2} + \cdots ,

the sum of which has to be found as a function of $x$ and $w$ ; similarly in the other two cases.

Problem 3.220

(1)(2)(3)(4)

Let $s$ be a constant of sufficiently small modulus. Then the following formulas are valid for $F(z)=e^{sz}$ :

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(1)

F(z) = F(0) + \frac{z}{1!}\Delta F(0) + \frac{z(z-1)}{2!}\Delta ^{2}F(0) + \cdots + \frac{z(z-1)\cdots (z-n+1)}{n!}\Delta ^{n}F(0) + \cdots ;

(2)

F'(z) = \Delta F(z) - \frac{1}{2}\Delta ^{2}F(z) + \frac{1}{3}\Delta ^{3}F(z) - \cdots + (-1)^{n-1}\frac{1}{n}\Delta ^{n}F(z) + \cdots ;

(3)

F(z) = F(0) + \frac{z}{1!}F'(1) + \frac{z(z-2)}{2!}F''(2) + \cdots + \frac{z(z-n)^{n-1}}{n!}F^{(n)}(n) + \cdots \quad [\text{problem 210}];

(4)

F(z) = F(0) + \sum _{n=1}^{\infty } \frac{z^{2}(z^{2}-1^{2})(z^{2}-2^{2})\cdots [z^{2}-(n-1)^{2}]}{(2n)!}\Delta ^{2n}F(-n)

+ \sum _{n=1}^{\infty } \frac{z(z^{2}-1^{2})(z^{2}-2^{2})\cdots [z^{2}-(n-1)^{2}]}{(2n-1)!}\frac{\Delta ^{2n-2}[F(-n+2)-F(-n)]}{2} \quad [\text{problems 212, 216}].

Problem 3.221

The four formulas mentioned in problem 220 hold for any polynomial $F(z)$ . (In this case the series are obviously finite.)

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Problem 3.222

The formulas (1), (2) given in problem 220 are also valid for any rational function $F(z)$ if the real part of $z$ is larger than the real part of any of the finite poles of $F(z)$ ; formula (1) requires the additional condition that $F(z)$ be regular for $z=0,1,2,3,\ldots$ . --Do the formulas (3) and (4) hold for rational but not entire functions?

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Problem 3.223

(1-z)^{n}\sum _{k=-\infty }^{\infty } a_{k}z^{k} = \sum _{k=-\infty }^{\infty } \Delta ^{n}a_{k}z^{n+k}

where $a_{k}$ , $k=0,\pm 1,\pm 2,\ldots$ , denote arbitrary constants.

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Problem 3.224

Define

F(z) = a_{0}+a_{1}z+a_{2}z^{2}+\cdots +a_{n}z^{n}+\cdots

and establish the relation

\frac{1}{1+t}F\left(\frac{t}{1+t}\right) = a_{0} + \Delta a_{0}t + \Delta ^{2}a_{0}t^{2} + \cdots + \Delta ^{n}a_{0}t^{n} + \cdots .

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Problem 3.225

Define

F(z) = a_{0}+2a_{1}z+2a_{2}z^{2}+\cdots +2a_{n}z^{n}+\cdots \quad a_{-n}=a_{n}

and show that

\frac{1}{\sqrt{1+4t}}F\left(\frac{1+2t-\sqrt{1+4t}}{2t}\right) = a_{0} + \Delta ^{2}a_{-1}t + \Delta ^{4}a_{-2}t^{2} + \cdots + \Delta ^{2n}a_{-n}t^{n} + \cdots .

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Problem 3.226

Define

F(z) = 2a_{1}z+2a_{2}z^{2}+2a_{3}z^{3}+\cdots +2a_{n}z^{n}+\cdots , \quad a_{-n}=-a_{n}

and show that

\frac{1}{t}F\left(\frac{1+2t-\sqrt{1+4t}}{2t}\right) = a_{1}-a_{-1} + (\Delta ^{2}a_{0}-\Delta ^{2}a_{-2})t

+ (\Delta ^{4}a_{-1}-\Delta ^{4}a_{-3})t^{2} + \cdots + (\Delta ^{2n}a_{-n+1}-\Delta ^{2n}a_{-n-1})t^{n} + \cdots .

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Problem 3.227

\frac{\sin \pi x}{\pi } = x\left(1-\frac{x}{1}\right)\left(1+\frac{x}{1}\right)\left(1-\frac{x}{2}\right)\left(1+\frac{x}{2}\right)\cdots

\frac{\sin \pi x}{\pi x(1-x)} = \prod _{n=1}^{\infty } \left(1+\frac{x}{n}\right)\left(1-\frac{x}{n+1}\right) = \prod _{n=1}^{\infty } \left(1+\frac{x(1-x)}{n(n+1)}\right).

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Problem 3.228

$\sin \pi z$ is a single-valued function of $w=z(1-z)$ . The expansion of $\sin \pi z$ in powers of $w$ contains only positive coefficients (except the constant term) [problem 227].

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Problem 3.229

Prove that

\left[\frac{d^{n}(\pi -x)^{-n-1}\cos x}{dx^{n}}\right]_{x=0} > 0, \quad n=0,1,2,\ldots

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