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I know the McLaurin series expansion for the hyperbolic Sine and Cosine functions; they are defined as

$$\sinh x =\left(x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots \right)=\sum_{n=0}^{\infty} \frac{x^{2n +1}}{\left( 2n +1\right)!,} \tag{1}$$ $$ \cosh x =\left(1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots \right)=\sum_{n=0}^{\infty} \frac{x^{2n}}{\left( 2n \right)!,}, \tag{2}$$

My question: is there some definition for these series?: $$?=\left(x + \frac{x^3 c}{3!} + \frac{x^5 c^2}{5!} + \frac{x^7 c^3}{7!} + \cdots \right) =\sum_{n=0}^{\infty} \frac{x^{2n +1}c^n}{\left( 2n +1\right)!}, \tag{3}$$ $$?=\left(1 + \frac{x^2 c}{2!} + \frac{x^4 c^2}{4!} + \frac{x^6 c^3}{6!} + \cdots \right)=\sum_{n=0}^{\infty} \frac{x^{2n} c^n}{\left( 2n \right)!}. \tag{4}$$

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  • $\begingroup$ Is $ c $ a positive constant ? $\endgroup$ Commented Nov 20, 2020 at 20:22
  • $\begingroup$ yes, is a positive constant. $\endgroup$ Commented Nov 20, 2020 at 20:24
  • $\begingroup$ "Maclaurin", please $\endgroup$ Commented Nov 20, 2020 at 20:35

2 Answers 2

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If $c>0$, we have $$ \sum_{n=0}^{\infty} \frac{x^{2n +1}c^n}{\left( 2n +1\right)!} =\frac1{\sqrt{c}}\sum_{n=0}^{\infty}\frac{(x\sqrt{c})^{2n +1}}{\left( 2n +1\right)!} =\frac{\sinh(x\sqrt{c})}{\sqrt{c}}, $$ and similarly $$ \sum_{n=0}^{\infty} \frac{x^{2n} c^n}{\left( 2n \right)!} = \cosh(x\sqrt{c}). $$ On the other hand, if $c<0$, then we have $$ \sum_{n=0}^{\infty} \frac{x^{2n +1}c^n}{\left( 2n +1\right)!} =\frac1{\sqrt{-c}}\sum_{n=0}^{\infty}(-1)^n\frac{(x\sqrt{-c})^{2n +1}}{\left( 2n +1\right)!} =\frac{\sin(x\sqrt{-c})}{\sqrt{-c}}, $$ and similarly $$ \sum_{n=0}^{\infty} \frac{x^{2n} c^n}{\left( 2n \right)!} = \cos(x\sqrt{-c}). $$

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hint

If $ c>0 $ then

$$x^{2n}c^n=(x\sqrt{c})^{2n}$$

and

$$x^{2n+1}c^n=\frac{1}{\sqrt{c}}(x\sqrt{c})^{2n+1}$$

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