Bulk renaming of files in PowerShell with sequential numeric suffixes

The syntax is way off. A few issues:

• I'm not sure what $($.19981016) was intended to produce, but $( ) evaluates the expression in the parentheses and interpolates the result into the string, and you're getting the error because $.19981016is not a valid expression. The error would be repeated for each .jpg file in the directory.
• {0:00000000#} in a formatted string will create a zero-padded number of 9 digits, but a shorter way to do that is {0:D9}. However, I thought you wanted the zero-padded number to have 4 digits, so that should be {0:0000#} or {0:D4}.
• I'm not sure what Foreach {$i=1} { [...] is supposed to do. The keyword foreach can mean a foreach loop, or it can be shorthand for Foreach-Object. In this context, receiving input from a pipeline, it's necessarily the latter, but this syntax is incorrect either way.

This will do what you want, if I understand the description correctly:

$i = 1
Get-ChildItem *.jpg | %{Rename-Item $_ -NewName ('19981016_{0:D4}.jpg' -f $i++)}

The filenames will be 19981016_0001.jpg, 19981016_0002.jpg, 19981016_0003.jpg, etc.

A few notes:

• You said that you want filenames like 19981016_0001, but I'm assuming you want to keep the .jpg extension.
• You need to initialize $i, otherwise it will start from 0 if it's not yet defined, or worse yet, from some other number if $i was used previously in the same PowerShell session. For example, if you have 1,432 .jpg files in the directory, and you run the command first with -WhatIf, and then run it for real, the numbers will start from 1432.
• $i++ increments $i by 1. However, if you're using it as a value, it increments after the value is read; that's why if $i is undefined, it will start from 0.
• Get-ChildItem is the same as ls. I used the native PowerShell name, but they're interchangeable (ls, dir, and gci are all aliases for Get-ChildItem).
• %{ } is shorthand for Foreach-Object