The phi-function family by everywhere-convergent series

Use the first-variable-linear standard inner product on ${\mathbb{C}}^n$ and its norm $\mathrm{\parallel }x{\mathrm{\parallel }}_2=\sqrt{{\sum }_{i<n}\mathrm{\mid }{x}_i{\mathrm{\mid }}^2}$ from The standard formulas $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and ${\sum }_{k<n}{x}_k\overline{y_k}$ on ${\mathbb{C}}^n$ are inner products and The norm $\parallel v\parallel =\sqrt{⟨v,v⟩}$ induced by a real or complex inner product. The matrix norm $\mathrm{\parallel }X{\mathrm{\parallel }}_2$ is the operator norm of the linear map $x\mapsto Xx$ in The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||. On ${\mathbb{C}}^0=\{0\}$ this operator norm is $0$, as specified in that definition.

For each integer $k\ge 0$, define the scalar phi function and its finite-matrix action by

${\phi }_k(z):={\sum }_{j=0}^{\mathrm{\infty }}\frac{z^j}{(j+k)!},{\phi }_k(X):={\sum }_{j=0}^{\mathrm{\infty }}\frac{X^j}{(j+k)!}\mathrm{.}$

Here $z\in \mathbb{C}$, $X\in M_n(\mathbb{C})$, $n\ge 0$, $X^0=I_n$, and factorials are the usual positive integer factorials. The scalar and matrix phi series converge absolutely ↗ establishes that both series converge absolutely for every such input, so these expressions define values on their full stated domains. By the published power-series definition The complex exponential by its power series, ${\phi }_0(z)=\exp z$; for matrices, ${\phi }_0(X)$ is the bounded-operator exponential series of The exponential series of a bounded operator.

At zero, the series gives ${\phi }_k(0)=1\mathrm{/}k!$ and ${\phi }_k(0_{n\times n})=I_n\mathrm{/}k!$ for every $n\ge 0$. For $n\ge 1$, only the $j=0$ matrix term can be nonzero. For $n=0$, The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations and Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes identify $I_0$ with the unique empty matrix $0_{0\times 0}$; every power $X^j$ and every partial sum is that matrix. Thus ${\phi }_k(0_{0\times 0})=I_0\mathrm{/}k!=0_{0\times 0}$, including $k=0$.

The recurrence is always interpreted as a series identity, including at zero; division by $z$ is not part of this definition.